On Additive Higher Chow Groups of Affine Schemes · 2016-01-23 · On Additive Higher Chow Groups of Affine Schemes Amalendu Krishna and Jinhyun Park Received: July 5, 2015 Revised:

Documenta Math. 49

On Additive Higher Chow Groups of Affine Schemes

Amalendu Krishna and Jinhyun Park

Received: July 5, 2015

Revised: December 18, 2015

Communicated by Takeshi Saito

Abstract. We show that the multivariate additive higher Chowgroups of a smooth affine k-scheme Spec (R) essentially of finite typeover a perfect field k of characteristic 6= 2 form a differential gradedmodule over the big de Rham-Witt complex WmΩ•

R. In the univariatecase, we show that additive higher Chow groups of Spec (R) forma Witt-complex over R. We use these structures to prove an etaledescent for multivariate additive higher Chow groups.

2010 Mathematics Subject Classification: Primary 14C25; Secondary13F35, 19E15Keywords and Phrases: algebraic cycle, additive higher Chow group,Witt vectors, de Rham-Witt complex

1. Introduction

The additive higher Chow groups TCHq(X,n;m) emerged originally in [5] inpart as an attempt to understand certain relative higher algebraic K-groups ofschemes in terms of algebraic cycles. Since then, several papers [16], [17], [18],[19], [26], [27], [28] have studied various aspects of these groups. But lack ofa suitable moving lemma for smooth affine varieties has been a hindrance instudies of their local behaviors. Its projective sibling was known by [17]. Duringthe period of stagnation, the subject has evolved into the notion of ‘cycleswith modulus’ CHq(X |D,n) by Binda-Kerz-Saito in [1], [15] associated to pairs(X,D) of schemes and effective Cartier divisors D, setting a more flexibleground, while this desired moving lemma for the affine case was obtained byW. Kai [14] (See Theorem 4.1).The above developments now propel the authors to continue their programof realizing the relative K-theory Kn(X × Spec k[t]/(tm+1), (t)) in terms ofadditive higher Chow groups. More specifically, one of the aims in the programconsidered in this paper is to understand via additive higher Chow groups,the part of the above relative K-groups which was proven in [2] to give the

Documenta Mathematica 21 (2016) 49–89

50 Amalendu Krishna and Jinhyun Park

crystalline cohomology. This part turned out to be isomorphic to the de Rham-Witt complexes as seen in [12]. This article is the first of the authors’ papersthat relate the additive higher Chow groups to the big de Rham-Witt complexesWmΩ•

R of [8] and to the crystalline cohomology theory. This gives a motivicdescription of the latter two objects.While the general notion of cycles with modulus for (X,D) provides a widerpicture, the additive higher Chow groups still have a non-trivial operationnot shared by the general case. One such is an analogue of the Pontryaginproduct on homology groups of Lie groups, which turns the additive higherChow groups into a differential graded algebra (DGA). This product is inducedby the structure of algebraic groups on A1 andGm and their action onX×Ar =:X [r] for r ≥ 1.The usefulness of such a product was already observed in the earliest paperson additive 0-cycles by Bloch-Esnault [5] and Rulling [28]. This product onhigher dimensional additive higher Chow cycles was given in [19] for smoothprojective varieties. In §5 of this paper, we extend this product structure intwo directions: (1) toward multivariate additive higher Chow groups and (2)on smooth affine varieties. In doing so, we generalize some of the necessarytools, such as the following normalization theorem, proven as Theorem 3.2.Necessary definitions are recalled in §2.

Theorem 1.1. Let X be a smooth scheme which is either quasi-projective oressentially of finite type over a field k. Let D be an effective Cartier divisor onX. Then each cycle class in CHq(X |D,n) has a representative, all of whosecodimension 1 faces are trivial.

The above theorem for ordinary higher Chow groups was proven by Bloch andhas been a useful tool in dealing with algebraic cycles. In this paper, we usethe above theorem to construct the following structure of differential gradedalgebra and differential graded modules on the multivariate additive higherChow groups, where Theorem 1.2 is proven in Theorems 7.1, 7.10, and 7.11,while Theorem 1.3 is proven in Theorem 6.13.

Theorem 1.2. Let X be a smooth scheme which is either affine essentially offinite type or projective over a perfect field k of characteristic 6= 2

(1) The additive higher Chow groups TCHq(X,n;m)q,n,m∈N has a func-torial structure of a restricted Witt-complex over k.

(2) If X = Spec (R) is affine, then TCHq(X,n;m)q,n,m∈N has a structureof a restricted Witt-complex over R.

(3) For X as in (2), there is a natural map of restricted Witt-complexesτRn,m : WmΩn−1

R → TCHn(R, n;m).

Theorem 1.3. Let r ≥ 1. For a smooth scheme X which is either affine essen-tially of finite type or projective over a perfect field k of characteristic 6= 2, themultivariate additive higher Chow groups CHq(X [r]|Dm, n)q,n≥0 with modu-lus m = (m1, · · · ,mr), where mi ≥ 1, form a differential graded module over


On Additive Higher Chow Groups of Affine Schemes 51

the DGA TCHq(X,n; |m| − 1)q,n≥1, where |m| =∑ri=1mi. In particular,

each CHq(X [r]|Dm, n) is a W(|m|−1)(R)-module, when X = Spec (R) is affine.

The above structures on the univariate and multivariate additive higher Chowgroups suggest an expectation that these groups may describe the algebraicK-theory relative to nilpotent thickenings of the coordinate axes in an affinespace over a smooth scheme. The calculations of such relative K-theory byHesselholt in [9] and [10] show that any potential motivic cohomology whichdescribes the above relative K-theory may have such a structure.As part of our program of connecting the additive higher Chow groups with therelative K-theory, we show in [22] that the above map τRn,m is an isomorphismwhen X is semi-local in addition, and we show how one deduces crystallinecohomology from additive higher Chow groups. The results of this paper forma crucial part in the process.Recall that the higher Chow groups of Bloch and algebraicK-theory do not sat-isfy etale descent with integral coefficients. As an application of Theorem 1.3,we show that the etale descent is actually true for the multivariate additivehigher Chow groups in the following setting:

Theorem 1.4. Let r ≥ 1 and let X be a smooth scheme which is either affineessentially of finite type or projective over a perfect field k of characteristic 6= 2.Let G be a finite group of order prime to char(k), acting freely on X with thequotient f : X → X/G. Then for all q, n ≥ 0 and and m = (m1, · · · ,mr) withmi ≥ 1 for 1 ≤ i ≤ r, the pull-back map f∗ induces an isomorphism

CHq(X/G[r]|Dm, n)≃−→ H0(G,CHq(X [r]|Dm, n)).

Note that the quotient X/G exists under the hypothesis on X . Since thecorresponding descent is not yet known for the relative K-theory of nilpotentthickenings of the coordinate axes in an affine space over a smooth scheme, theabove theorem suggests that this descent could be indeed true for the relativeK-theory.

Conventions. In this paper, k will denote the base field which will be assumedto be perfect after §4. A k-scheme is a separated scheme of finite type overk. A k-variety is a reduced k-scheme. The product X × Y means usuallyX ×k Y , unless said otherwise. We let Schk be the category of k-schemes,Smk of smooth k-schemes, and SmAffk of smooth affine k-schemes. A schemeessentially of finite type is a scheme obtained by localizing at a finite subset(including ∅) of a finite type k-scheme. For C = Schk,Smk,SmAffk, we letCess be the extension of the category C obtained by localizing at a finite subset(including ∅) of objects in C. We let SmLock be the category of smooth semi-local k-schemes essentially of finite type over k. So, SmAff ess

k = SmAffk ∪SmLock for the objects. When we say a semi-local k-scheme, we always meanone that is essentially of finite type over k. Let SmProjk be the category ofsmooth projective k-schemes.



2. Recollection of basic definitions

For P1 = Projk(k[s0, s1]), we let y = s1/s0 its coordinate. Let := P1\1. For

n ≥ 1, let (y1, · · · , yn) ∈ n be the coordinates. A face F ⊂

n means a closedsubscheme defined by the set of equations of the form yi1 = ǫ1, · · · , yis = ǫsfor an increasing sequence ij|1 ≤ j ≤ s ⊂ 1, · · · , n and ǫj ∈ 0,∞. We

allow s = 0, in which case F = n. Let := P

1. A face of nis the closure

of a face in n. For 1 ≤ i ≤ n, let F 1

n,i ⊂ nbe the closed subscheme given

by yi = 1. Let F 1n :=

∑ni=1 F

1n,i, which is the cycle associated to the closed

subscheme n\ n. Let 0 =

0:= Spec (k). Let ιn,i,ǫ :

n−1 → n be the

inclusion (y1, · · · , yn−1) 7→ (y1, · · · , yi−1, ǫ, yi, · · · , yn−1).

2.1. Cycles with modulus. Let X ∈ Schessk . Recall ([21, §2]) that for ef-

fective Cartier divisors D1 and D2 on X , we say D1 ≤ D2 if D1 + D = D2

for some effective Cartier divisor D on X . A scheme with an effective divisor(sed) is a pair (X,D), where X ∈ Schess

k and D an effective Cartier divisor. Amorphism f : (Y,E)→ (X,D) of seds is a morphism f : Y → X in Schess

k suchthat f∗(D) is defined as a Cartier divisor on Y and f∗(D) ≤ E. In particular,f−1(D) ⊂ E. If f : Y → X is a morphism of k-schemes, and (X,D) is a sedsuch that f−1(D) = ∅, then f : (Y, ∅)→ (X,D) is a morphism of seds.

Definition 2.1 ([1], [15]). Let (X,D) and (Y ,E) be schemes with effectivedivisors. Let Y = Y \E. Let V ⊂ X×Y be an integral closed subscheme withclosure V ⊂ X × Y . We say V has modulus D (relative to E) if ν∗V (D × Y ) ≤

ν∗V (X×E) on VN, where νV : V

N→ V → X×Y is the normalization followed

by the closed immersion.

Recall the following containment lemma from [21, Proposition 2.4] (see also [1,Lemma 2.1] and [17, Proposition 2.4]):

Proposition 2.2. Let (X,D) and (Y ,E) be schemes with effective divisorsand Y = Y \ E. If V ⊂ X × Y is a closed subscheme with modulus D relativeto E, then any closed subscheme W ⊂ V also has modulus D relative to E.

Definition 2.3 ([1], [15]). Let (X,D) be a scheme with an effective divisor.For s ∈ Z and n ≥ 0, let zs(X |D,n) be the free abelian group on integral closedsubschemes V ⊂ X×

n of dimension s+n satisfying the following conditions:

(1) (Face condition) for each face F ⊂ n, V intersects X × F properly.

(2) (Modulus condition) V has modulus D relative to F 1n on X ×

n.

We usually drop the phrase “relative to F 1n” for simplicity. A cycle in

zs(X |D,n) is called an admissible cycle with modulus D. One checks that(n 7→ zs(X |D,n)) is a cubical abelian group. In particular, the groupszs(X |D,n) form a complex with the boundary map ∂ =

∑ni=1(−1)

i(∂∞i − ∂0i ),

where ∂ǫi = ι∗n,i,ǫ.

Definition 2.4 ([1], [15]). The complex (zs(X |D, •), ∂) is the nonde-generate complex associated to (n 7→ zs(X |D,n)), i.e., zs(X |D,n) :=



zs(X |D,n)/zs(X |D,n)degn. The homology CHs(X |D,n) := Hn(zs(X |D, •))for n ≥ 0 is called higher Chow group of X with modulus D. If X is equidi-mensional of dimension d, for q ≥ 0, we write CHq(X |D,n) = CHd−q(X |D,n).

Here is a special case from [21]:

Definition 2.5. Let X ∈ Schessk . For r ≥ 1, let X [r] := X × Ar. When

(t1, · · · , tr) ∈ Ar are the coordinates, and m1, · · · ,mr ≥ 1 are integers, let Dm

be the divisor on X [r] given by the equation tm1

1 · · · tmrr = 0. The groups

CHq(X [r]|Dm, n) are calledmultivariate additive higher Chow groups ofX . Forsimplicity, we often say “a cycle with modulus m” for “a cycle with modulusDm.” For an r-tuple of integers m = (m1, · · · ,mr), we write |m| =

∑ri=1mi.

We shall say that m ≥ p if mi ≥ p for each i.When r = 1, we obtain additive higher Chow groups, and as in [19], we often usethe older notations Tzq(X,n+ 1;m− 1) for zq(X [1]|Dm, n) and TCHq(X,n+1;m − 1) for CHq(X [1]|Dm, n). In such cases, note that the modulus m isshifted by 1 from the above sense.

Definition 2.6. Let W be a finite set of locally closed subsets of X andlet e : W → Z≥0 be a set function. Let zqW,e(X |D,n) be the subgroup

generated by integral cycles Z ∈ zq(X |D,n) such that for each W ∈ Wand each face F ⊂

n, we have codimW×F (Z ∩ (W × F )) ≥ q − e(W ).They form a subcomplex zqW,e(X |D, •) of zq(X |D, •). Modding out by de-

generate cycles, we obtain the subcomplex zqW,e(X |D, •) ⊂ zq(X |D, •). We

write zqW(X |D, •) := zqW,0(X |D, •). For additive higher Chow cycles, we write

TzqW(X,n;m) for zqW[1](X [1]|Dm+1, n− 1), where W [1] = W [1] | W ∈ W.

Here are some basic lemmas used in the paper:

Lemma 2.7 ([21, Lemma 2.2]). Let f : Y → X be a dominant map of normalintegral k-schemes. Let D be a Cartier divisor on X such that the genericpoints of Supp(D) are contained in f(Y ). Suppose that f∗(D) ≥ 0 on Y . ThenD ≥ 0 on X.

Lemma 2.8 ([21, Lemma 2.9]). Let f : Y → X be a proper morphism of quasi-projective k-varieties. Let D ⊂ X be an effective Cartier divisor such thatf(Y ) 6⊂ D. Let Z ∈ zq(Y |f∗(D), n) be an irreducible cycle. Let W = f(Z) onX ×

n. Then W ∈ zs(X |D,n), where s = codimX×n(W ).

Lemma 2.9. Let X be a k-scheme, and let Uii∈I be an open cover of X.Let Z ∈ zq(X ×

n) and let ZUi be the flat pull-back to Ui × n. Then

Z ∈ zq(X |D,n) if and only if for each i ∈ I, we have ZUi ∈ zq(Ui|DUi , n),where DUi is the restriction of D on Ui.

Proof. The direction (⇒) is obvious since flat pull-backs respect admissibilityof cycles with modulus by [21, Proposition 2.12]. For the direction (⇐), wemay assume Z is irreducible. In this case, it is easily checked that the face andthe modulus conditions are both local on the base X .

2.2. de Rham-Witt complexes.



2.2.1. Ring of big Witt-vectors. Let R be a commutative ring with unit. Werecall the definition of the ring of big Witt-vectors of R (see [11, §4] or [28,Appendix A]). A truncation set S ⊂ N is a non-empty subset such that ifs ∈ S and t|s, then t ∈ S. As a set, let WS(R) := RS and define the mapw : WS(R) → RS by sending a = (as)s∈S to w(a) = (w(a)s)s∈S , where

w(a)s :=∑t|s ta

s/tt . When RS on the target of w is given the component-wise

ring structure, it is known that there is a unique functorial ring structure onWS(R) such that w is a ring homomorphism (see [11, Proposition 1.2]). WhenS = 1, · · · ,m, we write Wm(R) := WS(R).There is another description. Let W(R) := WN(R). Consider the multiplicativegroup (1 + tR[[t]])×, where t is an indeterminate. Then there is a naturalbijection W(R) ≃ (1+tR[[t]])×, where the addition in W(R) corresponds to themultiplication of formal power series. For a truncation set S, we can describeWS(R) as the quotient of (1+ tR[[t]])

× by a suitable subgroup IS . See [28, A.7]for details. In case S = 1, · · · ,m, we can write Wm(R) = (1+ tR[[t]])×/(1+tm+1R[[t]])× as an additive group.For a ∈ R, the Teichmuller lift [a] ∈ WS(R) corresponds to the image of1 − at ∈ (1 + tR[[t]])×. This yields a multiplicative map [−] : R → WS(R).The additive identity element of Wm(R) corresponds to the unit polynomial 1and the multiplicative identity element corresponds to the polynomial 1− t.

2.2.2. de Rham-Witt complex. Let p be an odd prime and R be a Z(p)-algebra.1

For each truncation set S, there is a differential graded algebra WSΩ•R called

the big de Rham-Witt complex over R. This defines a contravariant functoron the category of truncation sets. This is an initial object in the category ofV -complexes and in the category of Witt-complexes over R. For details, see [8]and [28, §1]. When S is a finite truncation set, we haveWSΩ

•R = Ω•

WS(R)/Z/N•S ,

where N•S is the differential graded ideal given by some generators ([28, Propo-

sition 1.2]). In case S = 1, 2, · · · ,m, we write WmΩ•R for this object.

Here is another relevant object for this paper from [8, Definition 1.1.1];a restricted Witt-complex over R is a pro-system of differential graded Z-algebras ((Em)m∈N,R : Em+1 → Em), with homomorphisms of graded rings(Fr : Erm+r−1 → Em)m,r∈N called the Frobenius maps, and homomorphismsof graded groups (Vr : Em → Erm+r−1)m,r∈N called the Verschiebung maps,satisfying the following relations for all n, r, s ∈ N:

(i) RFr = FrRr,RrVr = VrR, F1 = V1 = Id, FrFs = Frs, VrVs = Vrs;

(ii) FrVr = r. When (r, s) = 1, FrVs = VsFr on Erm+r−1;(iii) Vr(Fr(x)y) = xVr(y) for all x ∈ Erm+r−1 and y ∈ Em; (projection

formula)(iv) FrdVr = d, where d is the differential of the DGAs.

Furthermore, we require that there is a homomorphism of pro-rings (λ :Wm(R)→ E0

m)m∈N that commutes with Fr and Vr, satisfying

1A definition of Witt-complex over a more general ring R can be found in [11, Defini-tion 4.1].



(v) Frdλ([a]) = λ([a]r−1)dλ([a]) for all a ∈ R and r ∈ N.

The pro-system WmΩ•Rm≥1 is the initial object in the category of restricted

Witt-complexes over R (See [28, Proposition 1.15]).

3. Normalization theorem

Let k be any field. The aim of this section is to prove Theorem 3.2. Suchresults were known when D = ∅, or when X is replaced by X × A1 withD = tm+1 = 0 for t ∈ A1. We generalize it to higher Chow groups withmodulus.

Definition 3.1. Let (X,D) be a scheme with an effective divisor. LetzqN (X |D,n) be the subgroup of cycles α ∈ zq(X |D,n) such that ∂0i (α) = 0 forall 1 ≤ i ≤ n and ∂∞i (α) = 0 for 2 ≤ i ≤ n. One checks that ∂∞1 ∂

∞1 = 0. Writ-

ing ∂∞1 as ∂N , we obtain a subcomplex ι : (zqN (X |D, •), ∂N) → (zq(X |D, •), ∂).

Theorem 3.2. Let X ∈ Smessk and let D ⊂ X be an effective Cartier divisor.

Then ι : zqN(X |D, •)→ zq(X |D, •) is a quasi-isomorphism. In particular, everycycle class in CHq(X |D,n) can be represented by a cycle α such that ∂ǫi (α) = 0for all 1 ≤ i ≤ n and ǫ = 0,∞.

Let Cube be the standard category of cubes (see [24, §1]) so that a cubicalabelian group is a functor Cube

op → (Ab). Recall also from loc.cit. that anextended cubical abelian is a functor ECube

op → (Ab), where ECube is thesmallest symmetric monoidal subcategory of Sets containing Cube and themorphism µ : 2→ 1. The essential point of the proof of Theorem 3.2 is

Theorem 3.3. Let X ∈ Smessk and D ⊂ X be an effective Cartier divisor.

Then (n 7→ zq(X |D;n)) is an extended cubical abelian group.

If Theorem 3.3 holds, then [24, Lemma 1.6] implies Theorem 3.2. We suppose(X,D) is as in Theorem 3.2 in what follows. The idea is similar to that of [19,Appendix].Let q1 : 2 → be the morphism (y1, y2) 7→ y1 + y2 − y1y2 if y1, y2 6= ∞,and (y1, y2) 7→ ∞ if y1 or y2 = ∞. Under the identification ψ : ≃ A1 givenby y 7→ 1/(1− y) (which sends ∞, 0 to 0, 1), this map q1 is equivalentto q1,ψ : A2 → A1 given by (y1, y2) 7→ y1y2. For our convenience, we usethis ψ := (A1, 0, 1) and cycles on X ×

nψ. The boundary operator is

∂ =∑ni=1(−1)

i(∂0i − ∂1i ), and we replace F 1

n,i by F∞n,i = yi = ∞. We write

F∞n =

∑ni=1 F

∞n,i. We write ψ = (P1, 0, 1). The group of admissible cycles is

zqψ(X |D,n). Consider qn,ψ : X×n+1ψ → X×

nψ given by (x, y1, · · · , yn+1) 7→

(x, y1, · · · , yn−1, ynyn+1).

Proposition 3.4. For Z ∈ zqψ(X |D,n), we have q∗n,ψ(Z) ∈ zqψ(X |D,n+ 1).

The delicacy of its proof lies in that the product map q1,ψ : A2 → A1 does notextend to a morphism (P1)2 → P

1 of varieties so that checking the moduluscondition becomes nontrivial. We use a correspondence instead. For n ≥ 1, let



in : Wn → X × n+1ψ ×

1

ψ be the closed subscheme defined by the equation

u0ynyn+1 = u1, where (y1, · · · , yn+1) ∈ n+1ψ and (u0;u1) ∈

1ψ are the coordi-

nates. Let y := u1/u0. Its Zariski closureWn → X×n+1

ψ ×1

ψ is given by theequation u0un,1un+1,1 = u1un,0un+1,0, where (u1,0, u1,1), · · · , (un+1,0, un+1,1)

are the homogeneous coordinates of n+1

ψ with yi = ui,1/ui,0.

Consider θn : X×n+1ψ ×

1ψ → X×

nψ given by (x, y1, · · · , yn+1, (u0;u1)) 7→

(x, y1, · · · , yn−1, ynyn+1), and let πn := θn|Wn . To extend this πn to a mor-

phism πn onWn, we use the projection θn : X×n+1

ψ ×1

ψ → X×n−1

ψ ×1

ψ,that drops the coordinates (un,0;un,1) and (un+1,0;un+1,1), and the projection

pn : X ×n+1ψ ×

1

ψ → X ×n+1ψ , that drops the last coordinate (u0;u1).

Lemma 3.5. (1) Wn ∩ u0 = 0 = ∅, so that Wn ⊂ X × n+1ψ ×

1ψ. (2)

θn|Wn = πn. Thus, we define πn := θn|Wn, which extends πn. (3) The varieties

Wn and Wn are smooth. (4) Both πn and πn are surjective flat morphisms ofrelative dimension 1.

Proof. Its proof is almost identical to that of [19, Lemma A.5]. Part (1) followsfrom the defining equation ofWn, and (2) holds by definition. Let ρn := pn|Wn :Wn → X×n+1

ψ . SinceX is smooth, using Jacobian criterion we check thatWn

is smooth. Furthermore, ρn is an isomorphism with the obvious inverse. Underthis identification, the morphism πn can also be regarded as the projection(x, y1, · · · , yn, y) 7→ (x, y1, · · · , yn−1, y) that drops yn. In particular, πn is asmooth and surjective of relative dimension 1. To check that Wn is smooth,one can do it locally on each open set where each of un,i, un+1,i, ui is nonzero for

i = 0, 1. In each such open set, the equation for Wn takes the same form as forWn, so that it is smooth again by Jacobian criterion. Similarly as for πn, onesees πn is of relative dimension 1. Since θn is projective and πn is surjective,the morphism πn is projective and surjective. So, sinceWn is smooth, the mapπn is flat by [7, Exercise III-10.9, p.276]. Thus, we have (3) and (4).

Lemma 3.6. Let n ≥ 1 and let Z ⊂ X×nψ be a closed subscheme with modulus

D. Then Z ′ := (in)∗(π∗n(Z)) also has modulus D.

Proof. Let Z and Z′be the Zariski closures of Z and Z ′ in X ×

nψ and

X ×n+1

ψ , respectively. By Lemma 3.5 and the projectivity of θn, we see that

θn(Z′) = Z. Consider the commutative diagram

(3.1) Z′N

f

g//

νZ′

++

Wn in //

πn

%%

X ×n+1ψ ×

1ψ

θn

ZN νZ // X ×

nψ,



where f is induced by the surjection θn|Z′ : Z′→ Z, the maps g and νZ

are normalizations of Z′and Z composed with the closed immersions, and

νZ′ := in g. By the definition of θn, we have θ∗

n(D × n

ψ) = D × n+2

ψ ,

θ∗

n(F∞n,n) = F∞

n+2,n+2, while θ∗

n(F∞n,i) = F∞

n+2,i for 1 ≤ i ≤ n − 1. By the

defining equation of Wn, we have π∗nF

∞n,n = i

∗nF

∞n+2,n+2 = i

∗nu0 = 0 ≤

i∗n(un,0 = 0+ un+1,0 = 0) = i

∗n(F

∞n+2,n + F∞

n+2,n+1).

Thus, ν∗Z′θ∗

n

∑ni=1 F

∞n,i =

∑n−1i=1 ν

∗Z′F∞

n+2,i + g∗π∗nF

∞n,n ≤

∑n−1i=1 ν

∗Z′F∞

n+2,i +

g∗i∗n(F

∞n+2,n+F∞

n+2,n+1) =∑n+1

i=1 ν∗Z′F∞

n+2,i ≤∑n+2

i=1 ν∗Z′F∞

n+2,i. (In case n = 1,

we just ignore the terms with∑n−1

i=1 in the above.)

That Z has modulus D means ν∗Z(D × n

ψ) ≤∑n

i=1 ν∗ZF

∞n,i. Applying f∗ and

using (3.1), we have ν∗Z′(D×n+2

ψ ) = ν∗Z′θ∗

n(D×n

ψ) ≤ ν∗Z′θ

∗

n

∑ni=1 F

∞n,i, which

is bounded by∑n+2i=1 ν

∗Z′F∞

n+2,i as we saw above. This means Z ′ has modulusD.

Definition 3.7. For any closed subscheme Z ⊂ X ×nψ, we define Wn(Z) :=

pn∗in∗π∗n(Z), which is closed in X ×

n+1ψ .

Lemma 3.8. Let n ≥ 1. If a closed subscheme Z ⊂ X ×nψ intersects all faces

properly, then Wn(Z) intersects all faces of X ×n+1ψ properly.

Proof. OurWn is equal to τ∗τ∗nτ∗n+1W

Xn , whereWX

n is that of [23, Lemma 4.1],and τ, τn, τn+1 are the involutions (x 7→ 1− x) for y, yn, yn+1, respectively. So,the lemma is a special case of loc.cit.

Proof of Proposition 3.4. Consider the commutative diagram

Wn

πn

ρn=pn|Wn

''

in // X ×n+1ψ ×ψ

pn

X ×nψ X ×

n+1ψ .

qn,ψoo

By Lemma 3.5, ρn is an isomorphism so that ρn∗i∗np

∗n = Id. Hence, q∗n,ψ(Z) =

ρn∗i∗np

∗nq

∗n,ψ(Z) =

† ρn∗π∗n(Z) =

‡ pn∗in∗π∗n(Z) = Wn(Z), where †, ‡ are due to

commutativity. So, we have reduced to showing that Wn(Z) ∈ zqψ(X |D,n+1).

But, by Lemmas 3.6 and 3.8, we have in∗π∗n(Z) ∈ z

q+1ψ (X × P1|D× P1, n+ 1).

Now, for the projection pn, by Lemma 2.8, we have Wn(Z) = pn∗in∗π∗n(Z) ∈

zqψ(X |D,n+ 1). This proves Proposition 3.4.

Proof of Theorem 3.3. Since we know that (n 7→ zq(X |D;n)) is a cubicalabelian group, every morphism h : r → s in Cube induces a morphismh : r →

s which gives a homomorphism h∗ : zq(X |D, s) → zq(X |D, r).Furthermore, the morphism µ : 2 → 1 induces the morphism q1 : 2 →

1

of varieties, and for each Z ∈ zq(X |D, 1), we have q∗1(Z) ∈ zq(X |D, 2). In-deed, under the isomorphism ψ : ≃ A1, y 7→ 1/(1 − y), this is equivalent to



show that q∗1,ψ sends admissible cycles to admissible cycles, which we know byProposition 3.4.So, it only remains to show the following “stability under products”: if hi : ri →si, i = 1, 2, are morphisms in ECube such that the corresponding morphismshi : ri →

si induce homomorphisms h∗i : zq(X |D, si) → zq(X |D, ri), fori = 1, 2 and all q ≥ 0, then h := h1 × h2 :

r1+r2 → s1+s2 induces a

homomorphism h∗ : zq(X |D, s) → zq(X |D, r) for all q ≥ 0, where r = r1 + r2and s = s1 + s2.Since h = h1 × h2 = (Idr1 × h2) (h1 × Idr2), we reduce to prove it when his either Idr1 × h2 or h1 × Idr2 . But the statement obviously holds for thesecases.

4. On moving lemmas

Let k be any field. In this section, we discuss some of moving lemmas onalgebraic cycles with modulus conditions. By a ‘moving lemma’, we ask whetherthe inclusion zqW(Y |D, •) ⊂ zq(Y |D, •) in Definition 2.6 is a quasi-isomorphism.It is known when Y is smooth quasi-projective and D = 0 (by [4]), and whenY = X×A1, with X smooth projective, D = X ×tm+1 = 0, and W consistsof W ×A1 for finitely many locally closed subsets W ⊂ X (by [17]). Recently,W. Kai [14] proved it when Y is smooth affine with a suitable condition. Kai’scases include the above case of Y = X×A1, where X is this time smooth affine.His proof applies to more general cases, possibly after Nisnevich sheafifications.In §4.1, we sketch the argument of Kai in the case of multivariate additivehigher Chow groups of smooth affine k-variety. In §4.2, we generalize themoving lemma of [17] in the case of pairs (X × S,X ×D) where X is smoothprojective. In §4.3 and 4.4, we discuss the standard pull-back property and itsconsequences. In §4.5, we discuss a moving lemma for additive higher Chowgroups of smooth semi-local k-schemes essentially of finite type.

4.1. Kai’s affine method for multivariate additive higher Chow

groups. The moving lemma of W. Kai [14] is the first moving result thatapplies to cycle groups with a non-zero modulus over a smooth affine scheme.Since the work loc. cit. is at present not yet refereed, we give a detailed sketchthe proof of the following special case on multivariate additive higher Chowgroups. But, we emphasize that the most crucial part is due to Kai. FollowingDefinition 2.5, we write X [r] := X × Ar.

Theorem 4.1 (W. Kai). Let X be a smooth affine variety over any field k.Let W be a finite set of locally closed subsets of X. Let W [r] := W [r] | W ∈W. Let m = (m1, · · · ,mr) ≥ 1. Then the inclusion zqW[r](X [r]|Dm, •) →

zq(X [r]|Dm, •) is a quasi-isomorphism.

First recall some preparatory results:

Lemma 4.2 ([17, Lemma 4.5]). Let f : X → Y be a dominant morphism ofnormal varieties. Suppose that Y is integral with the generic point η ∈ Y , andlet Xη be the fiber over η, with the inclusion jη : Xη → X. Let D be a Weil



divisor on X such that j∗η(D) ≥ 0. Then there exists a non-empty open subset

U ⊂ Y such that j∗U (D) ≥ 0, where jU : f−1(U) → X is the inclusion.

The following generalizes [17, Proposition 4.7]:

Proposition 4.3 (Spreading lemma). Let k ⊂ K be a purely transcendentalextension. Let (X,D) be a smooth quasi-projective k-scheme with an effectiveCartier divisor, and let W be a finite collection of locally closed subsets of X.Let (XK , DK) and WK be the base changes via Spec (K) → Spec (k). LetpK/k : XK → Xk be the base change map. Then the pull-back map

p∗K/k :zq(X |D, •)

zqW(X |D, •)→

zq(XK |DK , •)

zqWK(XK |DK , •)

is injective on homology.

Proof. It is similar to [17, Proposition 4.7]. We sketch its proof for the reader’sconvenience. If k is finite, then we can use the standard pro-ℓ-extension ar-gument to reduce the proof to the case when k is infinite, which we assumefrom now. We may also assume that tr.degkK < ∞ and furthermore thattr.degkK = 1, by induction. So, we have K = k(A1

k).Suppose Z ∈ zq(X |D,n) is a cycle that satisfies ∂Z ∈ zqW(X |D,n − 1),and ZK = ∂(BK) + VK for some BK ∈ zq(XK |DK , n + 1) and VK ∈zqWK

(XK |DK , n). Consider the inclusion zq(XK |DK , •) → zq(XK , •). Then

there is a non-empty open U ′ ⊂ A1k such that BK = BU ′ |η, VK = VU ′ |η,

Z×U ′ = ∂(BU ′)+VU ′ for someBU ′ ∈ zq(X×U ′, n+1), VU ′ ∈ zqW×U ′(X×U ′, n),where η is the generic point of U ′. Let jη : X × η → X × U ′ be the inclusion,which is flat.Since BK , VK satisfy the modulus condition, we have j∗η(X×U

′×F 1n+1−D×U

′×

n+1

) ≥ 0 on BN

K and similarly for VN

K . Furthermore, BN

U ′ → U ′, VN

U ′ → U ′

are dominant. Thus by Lemma 4.2, there is a non-empty open U ⊂ U ′ such

that j∗U (X × U′ × F 1

n+1 − D × U′ ×

n+1) ≥ 0 on B

N

U and similarly for VN

U ,for jU : X × U → X × U ′. This proves that BU and VU have modulus D× U .Hence, BU ∈ zq(X × U |D × U, n+ 1) and VU ∈ z

qW×U(X × U |D × U, n) with

Z × U = ∂(BU ) + VU .Since k is infinite, the set U(k) → U is dense. We claim the following:Claim: There is a point u ∈ U(k) such that the pull-backs of BU and VUunder the inclusion iu : X×u → X×U are both defined in zq(X,n+1) andzqW(X,n), respectively.Its proof requires the following elementary fact:Lemma: Let Y be any k-scheme. Let B ∈ zq(Y × U) be a cycle. Then thereexists a nonempty open subset U ′′ ⊂ U such that for each u ∈ U ′′(k), theclosed subscheme Y × u intersects B properly on Y × U , thus it defines acycle i∗u(B) ∈ zq(Y ), where Y is identified with Y × u.Note that for each u ∈ U(k), the subscheme Y × u ⊂ Y × U is an effectivedivisor, so its proper intersection with B is equivalent to that Y × u doesnot contain any irreducible component of B. If there exists a point ui ∈ U(k)



such that Y × ui contains an irreducible component Bi of B, then for anyother u ∈ U(k) \ ui, we have (Y × u) ∩ Bi = ∅. So, for every irreduciblecomponent Bi of B, there exists at most one ui ∈ U(k) such that Y × uicontains Bi. Let S be the union of such points ui, if they exist. There are onlyfinitely many irreducible components of B, so |S| < ∞. Taking U ′′ := U \ S,we have Lemma.We now proveClaim. Let F ⊂

n+1 be any face, including the case F = n+1.

Since BU ∈ zq(X×U, n+1), by definition X×U×F and BU intersect properlyon X × U ×

n+1, so their intersection gives a cycle BU,F ∈ zq(X × U × F ).By Lemma with Y = X × F , there exists a nonempty open subset UF ⊂ Usuch that BU,F defines a cycle in zq(X × u × F ) for every u ∈ UF (k). LetU1 :=

⋂F UF , where the intersection is taken over all faces F of n+1. This is

a nonempty open subset of U . Similarly, let F ⊂ n be any face, including the

case F = n. Here, VU ∈ z

qW×U (X ×U, n), and repeating the above argument

involving Lemma with Y =W ×F forW ∈ W , we get a nonempty open subsetUW,F ⊂ U such that we have an induced cycle in zq(W × u × F ) for everyu ∈ UW,F (k). Let U2 :=

⋂W,F UW,F , where the intersection is taken over all

pairs (W,F ), with W ∈ W and a face F ⊂ n. Taking U := U1 ∩ U2, which is

a nonempty open subset of U , we now obtain Claim for every u ∈ U(k).Finally, for such a point u as in Claim, by the containment lemma (Proposition2.2), i∗u(BU ) and i

∗u(VU ) have modulus D. Hence, i∗u(BU ) ∈ z

q(X |D,n+1) andi∗u(VU ) ∈ z

qW(X |D,n). This finishes the proof.

Sketch of the proof of Theorem 4.1. Step 1. We first show it when X = Adk.Let K = k(Adk) and let η ∈ X be the generic point. To facilitate the proof,as we did previously in §3, using the automorphism y 7→ 1/(1 − y) of P1 wereplace (, ∞, 0) by (A1, 0, 1), and write = A1. We use the homogeneous

coordinates (ui,0;ui,1) ∈ 1= P1, where yi = ui,1/ui,0, then the divisor F 1

n,i in

the modulus condition is replaced by F∞n,i = yi =∞ and F

∞n =

∑ni=1 F

∞n,i.

For any g ∈ Ad and an integer s > 0, define φg,s : Adk(g)[r] ×k(g) 1k(g) →

Adk(g)[r] by φg,s(x, t, y) := (x + y(tm1

1 · · · tmrr )sg, t), where k(g) is the residue

field of g. (N.B. In terms of W. Kai’s homotopy, our g ∈ Ad corresponds to hisv = (g, 0, · · · , 0) ∈ Ad[r] = Ad+r.) For any cycle V ∈ zq(X [r]|Dm, n), define

H∗g,s(V ) := (φg,s×Idn)

∗p∗k(g)/k(V ), where pk(g)/k : Adk(g)[r]×n → Adk[r]×

n

is the base change.Using [3, Lemma 1.2], one checks that H∗

g,s(V ) preserves the face condition

for V . Moreover, if V ∈ zqW(X [r], n), then so does H∗g,s(V ). When g = η,

another application of [3, Lemma 1.2] shows that H∗g,s(V ) intersects with all

W [r] × F properly, where W ∈ W and a F ⊂ n is a face. The argument for

proving these face conditions follows the same steps as that of the proof of [17,Lemma 5.5, Case 2] though the present case is slightly different so that we use[3, Lemma 1.2] instead of [3, Lemma 1.1] (see [14, Lemma 3.5] for more detail).On the other hand, we have the following crucial and central assertion due toW. Kai (cf. [14, Proposition 3.3]):



Claim: For each irreducible V ∈ zq(Adk[r]|Dm, n), there is s(V ) ∈ Z≥0 such

that for any s > s(V ) and for any g ∈ Ad, the cycle H∗g,s(V ) has modulus Dm.

Once it is proven, call the smallest such integer s(V ), the threshold of V , forsimplicity. Here, instead of translations by g ∈ Ad used in usual higher Chowgroups of Ad (which correspond to s = 0), Kai uses adjusted translations as inthe definition of φg,s, so that near the divisors ti = 0, the effect of adjustedtranslation is also small, while away from the divisors ti = 0, the effect ofadjusted translation gets larger, so that for a sufficiently large s, this imbuesthe desired modulus condition into cycles. Note the following elementary fact

(cf. [14, Lemma 3.2]), which amounts to rewriting the definitions: Let A bea commutative ring with unity, p ⊂ A a prime ideal, ζ ∈ A, and u ∈ A \ p.Then the element ζ/u of κ(p) is integral over A/p if and only if there is ahomogeneous polynomial E(a, b) ∈ A[a, b], which is monic in the variable a,with E(ζ, b) ∈ p in A.

For each I ⊂ 1, · · · , n, consider the open subset UI ⊂ Adk × ngiven by the

conditions ui,0 6= 0 for i ∈ I and ui,1 6= 0 for i 6∈ I. For i 6∈ I, we let yi =

ui,0/ui,1 = y−1i . Hence, UI = Spec (RI), where RI := k[x, t, yii∈I , yii6∈I ],

where x = (x1, · · · , xd) and t = (t1, · · · , tr). On UI , the divisor F∞n used in

the definition of the modulus condition is given by the polynomial∏i6∈I yi.

For an irreducible V ∈ zq(Adk[r]|Dm, n), let V be its Zariski closure in Adk[r]×

n. For a given I, the restriction V ∩ (Adk[r] × UI) is given by an ideal of RI ,

say, generated by a finite set of polynomials f Iλ(x, t, yii∈I , yii6∈I) ∈ RI forλ ∈ ΛI .By the above fact and the assumption that V has the modulus condition,there is a polynomial EI(a, b) = EI(x, t, yii∈I , yii6∈I , a, b) ∈ RI [a, b], homo-geneous in a, b and monic in a, satisfying the condition inside the ring RI :

(4.1) EI(∏

i6∈I

yi, tm) ∈

∑

λ∈ΛI

(f Iλ),where tm = tm1

1 · · · tmrr .

If necessary, by multiplying a power of a to EI , we may assume degEI ≥degx f

Iλ, where deg is the homogeneous degree of EI in the variables a, b and

degx is the total degree with respect to x. In doing so, we may further assume

that degEI is the same for all subset I ⊂ 1, · · · , n. For this choice of degrees,we let s(V ) = degEI . If V is not irreducible, then take the maximum of s(Vi)over all irreducible components Vi of V to define s(V ). The heart of the proofis to show that this number satisfies the assertions of Claim, which we do now.We may assume V is irreducible. For any fixed s > s(V ) and g ∈ Ad, let V ′

be an irreducible component of H∗g,s(V ) and let V

′be its Zariski closure in

Adκ[r]×n+1

, where κ = k(g). We use the coordinates (y, y1, · · · , yn) ∈ n+1

,and for the first = P1, use the homogeneous coordinate (u0;u1) so that

y = u1/u0 and y := u0/u1 = y−1. Let ν : V′N→ V be the normalization.

Note that whether a divisor is effective or not on V′N

is a Zariski local question

on V′N

(thus on V′), so we may check the modulus condition Zariski locally



near any point P ∈ V′. Fix a point P . Let I ⊂ 1, · · · , n be the set points

i such that P does not map to ∞ ∈ P1κ of the (i + 1)-th projection V

′→

Adκ[r]× n+1→ κ = P1

κ.

There are two possibilities. In the first case P ∈ Adκ[r]×A1×

n, i.e. P does not

map to∞ ∈ P1 for the first projection to κ, the morphism pκ/k (φg,s×Idn) :

Adκ[r]×A1×n → Adk[r]×

n extends uniquely to Adκ[r]×A1×

n→ Adk[r]×

n.

Thus, by pulling-back the relation (4.1), we obtain in the ring RI [y],

EI(x+ y(tm)sg, t, yii∈I , yii6∈I ,∏

i6∈I

yi, tm) ∈(4.2)

∈∑

λ∈ΛI

(f Iλ(x+ y(tm)sg, t, yii∈I , yii6∈I)).

Here, the polynomials f Iλ(x+ y(tm)sg, yii∈I , yii6∈I) over λ ∈ ΛI define theunderlying closed subscheme of the Zariski closure of H∗

g,s(V ) restricted on the

region Spec (RI [y]). Due to the choice of the degrees of EI and fIλ , the relation

(4.2) implies that the rational function∏i6∈I yi/t

m is integral using fact. In

particular, V ′ satisfies the modulus condition in a neighborhood of P .In the remaining case P 6∈ Adκ[r] × A1 ×

n, i.e. P does map to ∞ ∈ P1

for the first projection to κ, we use the affine open chart Spec (RI [y]) where

u1 6= 0. The defining ideal of V′∩ Spec (RI [y]) in the ring RI [y] contains the

polynomials φIλ(x, t, y, yii∈I , yii6∈I) := f Iλ(x+ 1y (t

m)sg, t, yii∈I , yii6∈I) ·

ydegx(fIλ), where λ ∈ ΛI . By expanding the definition of φIλ, we see that it is of

the form

(4.3) φIλ = ydegx(fIλ)f Iλ(x, t, yii∈I , yii6∈I) + (tm)sh, h ∈ RI [y].

Express (4.1) as EI(∏i6∈I yi, t

m) =∑λ∈ΛI

bλfIλ for some bλ ∈ RI . Let cλ :=

ys(V )−degx(fIλ) · bλ (which is in RI because s(V ) ≥ degx(f

Iλ)). Then from (4.3),

(4.4)∑

λ∈ΛI

cλφIλ = ys(V ) · EI(

∏

i6∈I

yi, tm) + (tm)sg,

where (keep in mind that s ≥ s(V )) the right hand side becomes

(y∏i6∈I yi)

s(V ) + e1y(y∏i6∈I yi)

s(V )−1tm + · · · + (es(v)ys(V ) + (tm)s−s(V )h) ·

(tm)s(V ), which we write as E′(y∏i6∈I yi, t

m) for a polynomial E′(a, b) ∈

RI [y][a, b], homogeneous in a, b and monic in a. Thus (4.4) is∑

λ∈ΛIcλφ

Iλ =

E′(y∏i6∈I yi, t

m), which implies that the rational function y∏i6∈I yi/t

m is in-

tegral on V′∩ Spec (RI [y]) using fact. Thus V ′ also satisfies the modulus

condition near P . Combining these two cases, we have now proven Claim.Now consider the subgroup zqW[r],e(X [r]|Dm, n)

≤s ⊂ zqW[r],e(X [r]|Dm, n) for

s > 0, consisting of cycles V with its threshold s(V ) ≤ s (cf. [14, §3.4]). We



deducezqW[r],e(X [r]|Dm, n)

zqW[r](X [r]|Dm, n)= lim

→s

zqW[r],e(X [r]|Dm, n)≤s

zqW[r](X [r]|Dm, n)≤s.

Then one has the induced map from H∗η,s,

H∗η,s :

zqW[r],e(X [r]|Dm, n)≤s

zqW[r](X [r]|Dm, n)≤s→

zqW[r],e(XK [r]|Dm, n+ 1)

zqW[r](XK [r]|Dm, n+ 1),

which gives a homotopy between the base change p∗K/k and H∗η,s|y=1. How-

ever, H∗η,s|y=1 is zero on the quotient, while p∗K/k is injective on homology by

Proposition 4.3, after taking s → ∞, so that the map p∗K/k is in fact zero on

homology. This means, the quotient zqW[r],e(X [r]|Dm, n)/zqW[r](X [r]|Dm, n) is

acyclic, proving the moving lemma for X = Adk.Step 2. If X is a general smooth affine k-variety of dimension d, we usethe standard generic linear projection trick. We choose a closed immersionX → AN for some N ≫ d and run the steps of §6 of [17] (with Pn replaced byAN everywhere) mutatis mutandis to conclude the proof of the moving lemmafor X from that of affine spaces. We leave the details for the reader.

4.2. Projective method for multivariate additive higher Chow

groups. The following theorem generalizes the moving lemma for additivehigher Chow groups of smooth projective schemes [17, Theorem 4.1] to a gen-eral setting which includes the multivariate additive higher Chow groups.

Theorem 4.4. Let (S,D) be a smooth quasi-projective k-variety with an effec-tive Cartier divisor. Let X be a smooth projective k-variety. Let W be a finitecollection of locally closed subsets of X. We let W × S := W × S|W ∈W. Then the inclusion zqW×S(X × S|X × D, •) → zq(X × S|X × D, •)is a quasi-isomorphism. In particular, when m = (m1, · · · ,mr) ≥ 1, and(S,D) = (Ar, Dm), the moving lemma holds for multivariate additive higherChow groups of smooth projective varieties over k.

Proof. Most arguments of [17, Theorem 4.1] work with minor changes, so wesketch the proof.Step 1. We first prove the theorem when X = Pdk. The algebraic groupSLd+1,k acts on P

d. Let K = k(SLd+1,k). Then there is a K-morphismφ : 1

K → SLd+1,K such that φ(0) = Id, and φ(∞) = η, where η is the genericpoint of SLd+1,k. See [17, Lemma 5.4]. For such φ, consider the compositionHn of morphisms

Pd × S ×

n+1K

µφ→ P

d × S ×n+1K

pr′K→ Pd × S ×

nK

pK/k→ P

d × S ×nk ,

where µφ(x, s, y1, · · · , yn+1) = (φ(y1)x, s, y1, · · · , yn+1), pr′K is the projection

dropping y1, and pK/k is the base change. We claim that H∗n carries zqW×S(P

d×

S|Pd × D,n) to zqW×S(PdK × S|PdK × D,n + 1), i.e., for an irreducible cycle

Z ∈ zqW×S(Pd×S|Pd×S, n), we show that Z ′ := H∗

n(Z) ∈ zqW×S(P

dK ×S|P

dK×

D,n+ 1).



To do so, we first claim that Z ′ intersects with W × S × FK properly for eachW ∈ W and each face F ⊂

n+1.(1) In case F = 0 × F ′ for some face F ′ ⊂

n, because φ(0) = Id, we haveZ ′ ∩ (W × S × FK) ≃ ZK ∩ (W × S × F ′

K). Note that dim(W × S × FK) =dim(W ×S×F ′

K). Hence, codimW×S×FK (Z′∩ (W ×S×FK)) = dim(W ×S×

FK)−dim(Z ′∩(W×S×FK)) = dim(W×S×F ′K)−dim(ZK∩(W ×S×F ′

K)) =dim(W×S×F ′)−dim(Z∩(W×S×F ′)) = codimW×S×F ′(Z∩(W×S×F ′)) ≥ q,because Z ∈ zqW×S(P

d × S|Pd ×D,n).(2) In case F = ∞ × F ′ for some face F ′ ⊂

n, dim(W × S × FK) =dim(W×S×F ′

K) and Z ′∩(W×S×FK) ≃ η·(ZK)∩(W×S×F ′K), where SLd+1,k

acts on Pd×S×F ′, naturally on Pd and trivially on S×F ′. Let A :=W×S×F ′

and B := Z ∩ (Pd × S × F ′). Thus, codimW×S×FK (Z′ ∩ (W × S × FK)) =

dim(W×S×FK)−dim(Z ′∩(W×S×FK)) = dim(W×S×F ′K)−dim(η ·(ZK)∩

(W×S×F ′K)) =† dim(AK)−dim(η ·BK∩AK) = codimAK (η ·BK∩AK), where

† holds because Z ∩ A = B ∩ A. By applying [3, Lemma 1.1] to G = SLd+1,k,and the above A,B on X := Pd × S × F ′, there is a non-empty open subsetU ⊂ G such that for all g ∈ U , the intersection (g ·A) ∩B is proper on X . Byshrinking U , we may assume U is invariant under inverse map, so g = η−1 ∈ U .Thus, codimAK ((η ·BK)∩AK) ≥ codimXK (η ·BK). Since codimXK (η ·BK) =codimXKBK and codimXKBK = q, we get codimW×S×FK (Z

′∩(W×S×FK)) =codimAK ((η ·BK) ∩ AK) ≥ codimXKBK = q.(3) In case F = × F ′ for some face F ′ ⊂

n, the projection Z ′ ∩ (W ×S × × F ′

K) → K is flat, being a dominant map to a curve, so dim(Z ′ ∩(W × S × × F ′

K)) = dim(Z ′ ∩ (W × S × ∞ × F ′K)) + 1. We also have

dim(W × S × × F ′K) = dim(W × S × ∞ × F ′

K) + 1. Hence, we deducecodimW×S×FK (Z

′ ∩ (W ×S×FK)) = dim(W ×S××FK)− dim(Z ′ ∩ (W ×S ×× F ′

K)) = codimW×S×∞×F ′K(Z ′ ∩ (W × S × ∞ × F ′

K)) ≥† q, where

† follows from case (2). This shows Z ′ intersects all faces properly.Now we show that Z ′ has modulus P

d × D. We drop all exchange of thefactors, for simplicity. For p : Pd → Spec (k), we take V = p(Z) on S ×

n.Because Z ⊂ p−1(p(Z)) = Pr × V , we have Z ′ = µ∗

φ(1K × Z) ⊂ µ∗

φ(Pd ×

1K × V ) = Pd ×

1K × V := Z1. By Lemma 2.8, V is admissible on S ×

n.So, p∗[V ] = Pd × V is admissible on Pd × S ×

n. In particular, Pd × V hasmodulus Pd ×D. Hence, Z1 = Pd ×

1K × V also has modulus PdK ×D. Now,

Z ′ ⊂ Z1 shows that Z ′ has modulus PdK × D by Proposition 2.2. Thus, weproved Z ′ ∈ zqW×S(P

dK × S|P

dK ×D,n+ 1).

Going back to the proof, one checks that H∗• : zq(Pd×S|Pd×D, •)→ zq(PdK ×

S|Pd × D, • + 1) is a chain homotopy satisfying ∂H∗(Z) + H∗∂(Z) = ZK −η · (ZK), and the same holds for zW×S by a straightforward computation (see[17, Lemma 5.6]). Furthermore, for each admissible Z, we have η · ZK ∈zqWK×S(P

dK × S|P

dK × D,n), by the above proof of proper intersection of Z ′

with W × S × FK , where F = ∞ × F ′ for a face F ′ ⊂ n. Hence, the

base change p∗K/k : zq(Pdk × S|Pdk ×D, •)/z

qW×S(P

dk × S|P

dk ×D, •)→ zq(PdK ×

S|PdK × D, •)/zqWK×S(PdK × S|P

dK × D, •) is homotopic to η · p∗K/k, which is



zero on the quotient. That is, p∗K/k on the above quotient complex is zero

on homology. However, by the spreading argument (Proposition 4.3), p∗K/k is

injective on homology. (N.B. We used here an elementary fact that k(SLd+1,k)is purely transcendental over k. To check this fact, first note that by definitionk[SLd+1,k] ≃ k[Ti,j |1 ≤ i, j ≤ d + 1]/(det(M) − 1) for the (d + 1, d + 1)-matrix M = [Tij ] consisting of indeterminates Ti,j for 1 ≤ i, j ≤ d + 1. Hereby Cramer’s rule we can write det(M) − 1 = αTd+1,d+1 − β − 1, where α =det(Md+1,d+1), β =

∑1≤j≤d(−1)

d+1+j det(Md+1,j) and Mij is the (i, j)-minor

ofM . Here both α and β do not have Td+1,d+1. Hence k[SLd+1,k] ≃ k[Tij |1 ≤

i, j ≤ d+ 1, (i, j) 6= (d+ 1, d+ 1), β+1α ]. Thus, k(SLd+1,k) ≃ k(Tij|1 ≤ i, j ≤

d + 1, (i, j) 6= (d + 1, d + 1)), which is purely transcendental over k.) Hence,the quotient complex zq(Pd × S|Pd ×D, •)/zqW×S(P

d × S|Pd ×D, •) is acyclic,

i.e., the moving lemma holds for (Pd × S,Pd ×D), finishing Step 1.Step 2. Now let X be a general smooth projective variety of dimension d. Inthis case, we choose a closed immersion X → P

N for some N ≫ d. We now runthe linear projection argument of [17, §6] again without any extra argumentto deduce the proof of the moving lemma for X from that of the projectivespaces. We leave out the details.

4.3. Contravariant functoriality. The following contravariant functori-ality of multivariate additive higher Chow groups is an immediate applicationof the moving lemma and the proof is identical to that of [17, Theorem 7.1].

Theorem 4.5. Let f : X → Y be a morphism of k-varieties, with Y smoothaffine or smooth projective. Let r ≥ 1 and m = (m1, · · · ,mr) ≥ 1. Then thereexists a pull-back f∗ : CHq(Y [r]|Dm, n)→ CHq(X [r]|Dm, n).If g : Y → Z is another morphism with Z smooth affine or smooth projective,then we have (g f)∗ = f∗ g∗.

Remark 4.6. As a special case, when r = 1, we have the pull-back map f∗ :TCHq(Y, n;m)→ TCHq(X,n;m).

4.4. The presheaf T CH. For the rest of the section, we concentrate onadditive higher Chow groups. Let m ≥ 0. By Theorem 4.5, we see thatT qn,m := TCHq(−, n;m) is a presheaf of abelian groups on the categorySmAffk, but we do not know if it is a presheaf on the categories Smk orSchk. However, we can exploit Theorem 4.5 further to define a new presheafon Smk and Schk. The idea of this detour occurred to the authors whileworking on [20]. We do it for somewhat more general circumstances.Let C be a category and D be a full subcategory. Let F be a presheaf ofabelian groups on D, i.e. F : Dop → (Ab) is a functor to the category ofabelian groups. For each object X ∈ C, let (X ↓ D) be the category whoseobjects are the morphisms X → A in C, with A ∈ D, and a morphism fromh1 : X → A to h2 : X → B, with A,B ∈ D, is given by a morphism g : A→ Bin C such that g h1 = h2. The functor F : Dop → (Ab) induces the functor

(X ↓ D)op → (Ab) given by (Xh→ A) 7→ F (A), also denoted by F .



Definition 4.7. Suppose that for each X ∈ C, the category (X ↓ D) is cofil-tered. Then define F(X) := colim

(X↓D)opF .

In particular, when C = Schk and D = SmAffk, one checks that (X ↓SmAffk) is cofiltered, and for X ∈ Schk, we define T CHq(X,n;m) :=

colim(X↓SmAffk)op

T qn,m.

Proposition 4.8. Let C be a category and D be a full subcategory such that foreach X ∈ C, the category (X ↓ D) is cofiltered. Let F be a presheaf of abeliangroups on D and let F be as in Definition 4.7.Let f : X → Y be a morphism in C. Then for X ∈ C, the association X 7→F(X) satisfies the following properties:

(1) There is a canonical homomorphism αX : F(X)→ F (X).(2) If X ∈ D, then αX is an isomorphism, and α : F → F defines an

isomorphism of presheaves on D.(3) There is a canonical pull-back f∗ : F(Y )→ F(X). If g : Y → Z is an-

other morphism in C, then we have (gf)∗ = f∗g∗. So, F is a presheafof abelian groups on C. In particular, T CHq(−, n;m) is a presheafof abelian groups on Schk, which is isomorphic to TCHq(−, n;m) onSmAffk.

Proof. (1) Let (Xh−→ A) ∈ (X ↓ D)op. By the given assumption, we have

the pull-back h∗ : F (A) → F (X). Regarding F (X) as a constant functor on(X ↓ D)op, this gives a morphism of functors F → F (X). Taking the colimitsover all h, we obtain F(X)→ F (X), where αX = colimh h

∗.(2) When X ∈ D, the category (X ↓ D)op has the terminal object IdX : X →X . Hence, the colimit F(X) is just F (X).(3) A morphism f : X → Y in C defines a functor f ♯ : (Y ↓ D)op → (X ↓ D)op

given by (Yh→ A) 7→ (X

f→ Y

h→ A). Thus, taking the colimits of the functors

induced by F , we obtain f∗ : F(Y )→ F(X). For another morphism g : Y → Z,that (g f)∗ = f∗ g∗ can be checked easily using the universal property of thecolimits.In the special case when C = Schk and D = SmAffk with F = TCHq(−, n;m),by Theorem 4.5 we know that F is a presheaf on SmAffk. So, the above generaldiscussion holds.

Remark 4.9. Since additive higher Chow groups have pull-backs for flat maps(see [16, Lemma 4.7]), it follows that for X ∈ Smk, α(−) defines a map ofpresheaves T CHq(−, n;m)→ TCHq(−, n;m) on the small Zariski site XZar ofX . Proposition 4.8(2) says that this map is an isomorphism for affine opensubsets of X . Thus, this map of presheaves on XZar induces an isomorphismof their Zariski sheafifications.

4.5. Moving lemma for smooth semi-local schemes. One remaining ob-jective in Section 4 is to prove the following semi-local variation of Theorem4.1:



Theorem 4.10. Let Y ∈ SmLock. Let W be a finite set of locally closedsubsets of Y . Then the inclusion TzqW(Y, •;m) → Tzq(Y, •;m) is a quasi-isomorphism.

We begin with some basic results related to cycles over semi-local schemes.Recall that when A is a ring and Σ = p1, · · · , pN is a finite subset of Spec (A),

the localization at Σ is the localization A → S−1A, where S =⋂Ni=1(A \ pi).

For a quasi-projective k-scheme X and a finite subset Σ of (not necessarilyclosed) points of X , the localization XΣ is defined by reducing it to the casewhen X is affine by the following elementary fact (see [25, Proposition 3.3.36])that we use often.

Lemma 4.11. Let X be a quasi-projective k-scheme. Given any finite subsetΣ ⊂ X and an open subset U ⊂ X containing Σ, there exists an affine opensubset V ⊂ U containing Σ.

ForX ∈ Schk and a point x ∈ X , the open neighborhoods of x form a cofilteredcategory and we have functorial flat pull-back maps (jVU )∗ : Tzq(V, n;m) →Tzq(U, n;m) for jVU : U → V in this category.

Lemma 4.12. Let X ∈ Schk and let x ∈ X be a scheme point. Let Y =

Spec (OX,x). Then we have colimx∈U Tzq(U, n;m)≃−→ Tzq(Y, n;m), where the

colimit is taken over all open neighborhoods U of x.

Proof. Replacing x by an affine open neighborhood of x ∈ X , we may assumethat X is affine and write X = Spec (A). Let px ⊂ A be the prime ideal thatcorresponds to the point x and let S := A \ px, so that Y = Spec (S−1A). Tofacilitate our proof, using the automorphism y 7→ 1/(1 − y) of P1, we identify with A1 and take 0, 1 ⊂ A1 as the faces. So, X ×Bn = X ×A1 ×An−1 =Spec (A[t, y1, · · · , yn−1]).Let α ∈ Tzq(Y, n;m). We need to find an open subset U ⊂ X containing x suchthat the closure of α in U ×A1×An−1 is admissible. For this, we may assumeα is irreducible, i.e., it is a closed irreducible subscheme Z ⊂ Y × A1 × An−1.Let Z be its Zariski closure in X × A1 × An−1. Let p be the prime ideal ofB := A[t, y1, · · · , yn−1] such that V (p) = Z.For the proper intersection with faces, let q ⊂ B be the prime ideal (yi1 −ǫ1, · · · , yis − ǫs), where 1 ≤ i1 < · · · < is ≤ n− 1 and ǫj ∈ 0, 1. Let P be aminimal prime of p + q. One checks immediately from the behavior of primeideals under localizations that there is a ∈ S such that eitherPB[a−1] = B[a−1]or ht(PB[a−1]) ≥ q + s. This means, over Uq := Spec (A[a−1]), either the

intersection of ZUqwith V (q) is empty, or has codimension ≥ q + s. Applying

this argument to all faces, we can take U1 :=⋂

qUq. Then ZU1

intersects all

faces of U1 × A1 × An−1 properly.

For the modulus condition, let ν : ZN → Z → X × P1 × (P1)n−1 be the nor-malization composed with the closed immersion of the further Zariski closure

Z of Z. Let F∞n =

∑n−1i=1 yi = ∞ be the divisor at infinity. For an open

set j : U → X , the modulus condition of ZU means (m + 1)[j∗ν∗t = 0] ≤



[j∗ν∗(F∞n )] on ZNU . Note that there exist only finitely many prime Weil divi-

sors P1, · · · , Pℓ on ZN such that ordPi(ν∗(F∞

n )− (m+1)ν∗t = 0) < 0. Their

images Qi under the normalization map ZN → Z are still irreducible proper

closed subsets of Z, thus of X × P1 × (P1)n−1. Since Z = ZY has the modulus

condition on Y × Bn by the given assumption, we have (Y × Bn) ∩ Qi = ∅for each 1 ≤ i ≤ ℓ. Thus, there is an affine open subset U2 ⊂ X containing x

such that (U2 × Bn) ∩ Qi = ∅ for each 1 ≤ i ≤ ℓ. Now, by construction, ZU2

on U2 × Bn satisfies the modulus condition. So, taking an affine open subsetU ⊂ U1 ∩ U2 containing x, we have ZU ∈ Tzq(U, n;m). That (ZU )Y = Z isobvious.

We can extend this colimit description to semi-local schemes:

Lemma 4.13. Let Y be a semi-local k-scheme obtained by localizing at a finiteset Σ of scheme points of a quasi-projective k-variety X. For a cycle Z onY ×Bn, let Z be its Zariski closure in X ×Bn.Then Z ∈ Tzq(Y, n;m) if and only if there exists an affine open subset U ⊂ Xcontaining Σ, such that ZU ∈ Tzq(U, n;m), where ZU is the pull-back of Z viathe open immersion U → X.

Proof. The direction (⇐) is obvious by pulling back via the flat morphismY → U . For the direction (⇒), by Lemma 4.12, for each x ∈ Σ we have anaffine open neighborhood Ux ⊂ X of x such that ZUx ∈ Tzq(Ux, n;m). TakeW =

⋃x∈ΣUx. This is an open subset of X containing Σ. By Lemma 2.9, we

have ZW ∈ Tzq(W,n;m). On the other hand, by Lemma 4.11, there exists anaffine open subset U ⊂ W containing Σ. By taking the flat pull-back via theopen immersion U →W , we get ZU ∈ Tzq(U, n;m).

Lemma 4.14. Let Y be a semi-local integral k-scheme obtained by localizing ata finite set Σ of scheme points of an integral quasi-projective k-scheme X. LetZ ∈ Tzq(Y, n;m), W ∈ Tzq(Y, n+1;m), and let Z, W be their Zariski closuresin X × Bn and X × Bn+1, respectively. For every open subset U ⊂ X, thesubscript U means the pull-back to U . Then we have the following:

(1) If ∂Z = 0, we can find an affine open subset U ⊂ X containing Σ suchthat ZU ∈ Tzq(U, n;m) and ∂ZU = 0.

(2) If Z = ∂W , we can find an affine open subset U ⊂ X containing Σsuch that ZU ∈ Tzq(U, n;m), WU ∈ Tzq(U, n+1;m) and ZU = ∂WU .

Proof. Note that (1) is a special case of (2), so we prove (2) only. Let Z ′ :=Z − ∂W ∈ zq(X × Bn). If Z ′ is 0 as a cycle, then take U0 = X . If not,let Z ′

1, · · · , Z′s be the irreducible components of Z ′. Since Z = ∂W , each

component Z ′i has empty intersection with Y ×Bn. So, each π((Z ′

i)c) is a non-

empty open subset of X containing Σ, where π : X×Bn → X is the projection,which is open. Take U0 =

⋂si=1 π((Z

′i)c).

On the other hand, Lemma 4.13 implies that there exist open sets U1, U2 ⊂ Xcontaining Σ such that ZU1

∈ Tzq(U1, n;m) and WU2∈ Tzq(U2, n + 1;m).



Choose an affine open subset U ⊂ U0 ∩ U1 ∩ U2 containing Σ, using Lemma4.11. Then part (2) holds over U by construction.

Proof of Theorem 4.10. We show that the chain map TzqW(Y, •;m) →Tzq(Y, •;m) is a quasi-isomorphism. Let X be a smooth affine k-variety witha finite subset Σ ⊂ X such that Y = Spec (OX,Σ).For surjectivity on homology, let Z ∈ Tzq(Y, n;m) be such that ∂Z = 0.Let Z be the Zariski closure of Z in X × Bn. Here, ∂Z may not be zero,but by Lemma 4.14(1), there exists an affine open subset U ⊂ X containingΣ such that we have ∂ZU = 0, where ZU is the pull-back of Z to U . LetWU = WU |W ∈ W, where WU is the Zariski closure of W in U . Thenthe quasi-isomorphism TzqWU

(U, •;m) → Tzq(U, •;m) of Theorem 4.1 shows

that there are some C ∈ Tzq(U, n + 1;m) and Z ′U ∈ TzqWU

(U, n;m) such that

∂C = ZU − Z ′U . Let ι : Y → U be the inclusion. So, by applying the

flat pull-back ι∗ (which is equivariant with respect to taking faces), we obtain∂(ι∗C) = Z − ι∗Z ′

U , and here ι∗Z ′U ∈ TzqW(Y, n;m), i.e., Z is equivalent to a

member in TzqW(Y, n;m).For injectivity on homology, let Z ∈ TzqW(Y, n;m) be such that Z = ∂Z ′ for

some Z ′ ∈ Tzq(Y, n+ 1;m). Let Z and Z′be the Zariski closures of Z and Z ′

on X×Bn and X×Bn+1, respectively. Then by Lemma 4.14(2), there exists a

nonempty open affine subset U ⊂ X containing Σ such that ZU = ∂Z′

U . Thenthe quasi-isomorphism TzqWU

(U, •;m) → Tzq(U, •;m) of Theorem 4.1 shows

that there exists Z ′′ ∈ TzqWU(U, n + 1;m) such that ZU = ∂Z ′′. Pulling back

via ι : Y → U then shows Z = ∂(ι∗Z ′′), with ι∗Z ′′ ∈ TzqW(Y, n+ 1;m).

Using an argument identical to Theorem 4.5 (see [17, Theorem 7.1]), we get:

Corollary 4.15. Let f : Y1 → Y2 be a morphism in Schessk , where

Y2 ∈ SmLock. Then there is a natural pull-back f∗ : TCHq(Y2, n;m) →TCHq(Y1, n;m).

5. The Pontryagin product

Let R be a commutative ring and let (A, dA) be a differential graded algebraover R. Recall that (left) differential graded module M over A is a left A-module M with a grading M = ⊕n∈ZMn and a differential dM such thatAmMn ⊂ Mm+n, dM (Mn) ⊂ Mn+1 and dM (ax) = dA(a)x + (−1)nadM (x)for a ∈ An and x ∈ M . A homomorphism of differential graded modulesf :M → N over A is an A-module map which is compatible with gradings anddifferentials.In this section, we show that the multivariate additive higher Chow groupshave a product structure that resembles the Pontryagin product. We constructa differential operator on these groups in the next section and show that theproduct and the differential operator together turn multivariate additive higherChow groups groups into a differential graded module over WmΩ•

R for suitablem, when X = Spec (R) is in SmAffess

k . This generalizes the DGA-structure on



additive higher Chow groups of smooth projective varieties in [19]. The basefield k is perfect in this section.

5.1. Some cycle computations. We generalize some of [19, §3.2.1, 3.2.2,3.3]. Let (X,D) be a k-scheme with an effective divisor.Recall that a permutation σ ∈ Sn acts naturally on

n via σ(y1, · · · , yn) :=(yσ(1), · · · , yσ(n)). This action extends to cycles on X ×

n and X ×n.

Let n, r ≥ 1 be given. Consider the finite morphism χn,r : X ×n → X ×

n

given by (x, y1, · · · , yn) 7→ (x, yr1 , y2, · · · , yn). Given an irreducible cycle Z ⊂X ×

n, define Zr := (χn,r)∗([Z]) = [k(Z) : k(χn,r(Z))] · [χn,r(Z)]. Weextend it Z-linearly.

Lemma 5.1. If Z is an admissible cycle with modulus D, then so is Zr.

Proof. The proof is almost identical to that of [19, Lemma 3.11], except that

the divisor (m + 1)t = 0 there should be replaced by D × n. We give its

argument for the reader’s convenience.We may assume Z is irreducible. It is enough to show that χn,r(Z) is admissiblewith modulus D. We first check that it satisfies the face condition of Definition2.3. When n = 1, the proper faces of are of codimension 1, and for ǫ ∈0,∞, we have ∂ǫ1(χn,r(Z)) = r∂ǫ1(Z). When n ≥ 2, for ǫ ∈ 0,∞, we have∂ǫ1(χn,r(Z)) = r∂ǫ1(Z) and ∂ǫi (χn,r(Z)) = χn−1,r(∂

ǫi (Z)) if i ≥ 2. For faces

F ⊂ n of higher codimensions, we consequently have F ·(χn,r(Z)) = r(F ·Z) if

F involves the equations y1 = ǫ, and F ·(χn,r(Z)) = χn−c,r(F ·Z), otherwise,where c is the codimension of F . Since the intersection F · Z is proper, so isχn−c,r(F · Z) by induction on the codimension of faces. This shows χn,r(Z)satisfies the face condition.To show thatW := χn,r(Z) has modulus D, consider the commutative diagram

ZN νZ //

χNn,r

Z

χn,r

ιZ // X ×n

χn,r

WN νW // W

ιW // X ×n,

where Z, W are the Zariski closures of Z and W in X×nand νZ , νW are the

respective normalizations. The morphisms χn,r, χn,r are the natural induced

maps, and χNn,r is induced by the universal property of normalization. Since Zhas modulus D, we have the inequality

(5.1) [ν∗Zι∗Z(D ×

n)] ≤

n∑

i=1

[ν∗Zι∗Zyi = 1].

By the definition of χn,r, we have χ∗n,r(D ×

n) = D ×

n, χ∗

n,ry1 = 1 ≥y1 = 1, and χ∗

n,ryi = 1 = yi = 1 for i ≥ 2. Hence (5.1) implies that

[ν∗Zι∗Zχ

∗n,r(D ×

n)] ≤

∑ni=1[ν

∗Zι

∗Zχ

∗n,ryi = 1]. By the commutativity of the

diagram, this implies that χNn,r∗(∑n

i=1 ν∗W ι

∗W yi = 1 − ν∗W ι

∗W (D ×

n))≥ 0.



By Lemma 2.7, this implies∑ni=1 ν

∗W ι

∗W yi = 1− ν∗W ι

∗W (D×

n) ≥ 0, which

means W has modulus D. This completes the proof.

Let n, i ≥ 1. Suppose X is smooth quasi-projective essentially of finite type

over k. Let (x, y1, · · · , yn, y, λ) be the coordinates of X × n+2

. Consider theclosed subschemes V iX on X×n+2 given by the equation (1−y)(1−λ) = 1−y1if i = 1, and (1−y)(1−λ) = (1−y1)(1+y1+ · · ·+y

i−11 −λ(1+y1+ · · ·+y

i−21 ))

if i ≥ 2.

Let V iX be the Zariski closure of V iX in X × n+2

. Let π1 : X × n+2

→

X × n+1

be the projection that drops y1, and let π′1 := π1|V iX . As in [19,

Lemma 3.12], one sees that π′1 is proper surjective. For an irreducible cycle

Z ⊂ X × n, define (see [19, Definition 3.13]) γiZ := π′

1∗(ViX · (Z ×

2)) asan abstract algebraic cycle. One checks that it is also the Zariski closure ofνi(Z × ), where νi : X ×

n × → X × n+1 is the rational map given by

νi(x, y1, · · · , yn, y) = (x, y2, y3, · · · , yn, y,y−yi1y−yi−1

1

). We extend the definition of

γiZ Z-linearly.

Lemma 5.2. Let Z ∈ zq(X |D,n). Then γiZ ∈ zq(X |D,n+ 1).

Proof. Once we have Lemma 5.1, the proof of Lemma 5.2 is very similar to

that of [19, Lemma 3.15], except we replace (m+ 1)t = 0 by D ×n+1

. Wegive its argument for the reader’s convenience.We may assume Z is irreducible. To keep track of n, we write γiZ,n = γiZ . We

first check that it satisfies the face condition of Definition 2.3. Let ǫ ∈ 0,∞.Let F ⊂

n+1 be a face. If F involves the equation yj = ǫ for j = n, n+ 1,then by direction computations, we see that ∂0n(γ

iZ,n) = σ · Z, ∂0n+1(γ

iZ,n) =

σ · (Zi) for the cyclic permutation σ = (1, 2, · · · , n), and ∂∞n (γiZ,n) = 0,

∂∞n+1(γiZ,n) = σ · (Zi − 1). Since Z is admissible with modulus D, so are

Zi and Zi − 1 by Lemma 5.1. In particular, all of σ · Z, σ · (Zi), andσ · (Zi− 1) intersect all faces properly. Hence γiZ,n intersects F properly.

In case F does not involve the equations yj = ǫ for j = n, n + 1, we proveit by induction on n ≥ 1. By direction calculations, for j < n, we have∂ǫj(γ

iZ,n) = γi∂ǫjZ,n−1 so that the dimension of ∂ij(γ

iZ,n) is at least one less by

the induction hypothesis. Repeated applications of this argument for all otherdefining equations of F then give the result.It remains to show that γiZ has modulus D. Every irreducible component ofγiZ is of the form W ′ = π′

1(Z′), where Z ′ is an irreducible component of V iX ·

(Z × 2). We prove W ′ has modulus D. Consider the following commutative

diagram

Z′N

πN1

νZ′// V iX

ι //

π′1

π′1

$$

X ×n+2

π1

W′N ν //

W′ ιW ′

// X ×n+1

,



where νZ′ is the normalization of the Zariski closure Z′of Z ′ in V iX , ν is

the normalization of the Zariski closure W′of W ′ in X ×

n+1, and πN1 ,

π′1 are the induced morphisms. We use (x, y1, · · · , yn, y, λ) ∈ X ×

n+2and

(x, y2, · · · , yn, y, λ) ∈ X × n+1

as the coordinates. From the modulus Dcondition of Z, we deduce

(5.2) ν∗Z′ι∗(D ×n+2

) ≤n∑

j=1

ν∗Z′ι∗yj = 1.

Note that the above does not involve the divisors y = 1 and λ = 1. SinceV iX is an effective divisor on X ×

n+2 defined by the equation (1 − y1)(∗) =(1 − y)(1 − λ) for some polynomial (∗), we have [ν∗Z′ι∗y1 = 1] ≤ [ν∗Z′ι∗y =1] + [ν∗Z′ι∗λ = 1].

Since the above diagram commutes, from (5.2) we deduce πN1∗ν∗ι∗W ′ (D ×

n+1

) ≤ πN1∗(∑n

j=2 ν∗ι∗W ′yj = 1+ y = 1+ λ = 1

). Hence by Lemma

2.7, we deduce ν∗ι∗W ′ (D×n+1

) ≤∑n

j=2 ν∗ι∗W ′yj = 1+ y = 1+ λ = 1,

which means W ′ has modulus D. This finishes the proof.

Lemma 5.3. Let n ≥ 2 and let Z ∈ zq(X |D,n) such that ∂ǫi (Z) = 0 for all1 ≤ i ≤ n and ǫ ∈ 0,∞. Let σ ∈ Sn. Then there exists γσZ ∈ z

q(X |D,n+ 1)such that Z = (sgn(σ))(σ · Z) + ∂(γσZ).

Proof. Its proof is almost identical to that of [19, Lemma 3.16], except thatwe use Lemma 5.2 instead of [19, Lemma 3.15]. We give its argument for thereader’s convenience.First consider the case when σ is the transposition τ = (p, p + 1) for 1 ≤p ≤ n − 1. We do it for p = 1 only, i.e. τ = (1, 2). Other cases of τ aresimilar. Let ξ be the unique permutation such that ξ · (x, y1, · · · , yn+1) =(x, yn, y1, yn+1, y2, · · · , yn−1). Consider the cycle γτZ := ξ · γ1Z , where γ

1Z is as

in Lemma 5.2. Being a permutation of an admissible cycle, so is this cycle γξZ .Furthermore, by direction calculations, we have ∂∞1 (γτZ) = 0, ∂01(γ

τZ) = τ · Z,

∂∞3 (γτZ) = 0 and ∂03(γτZ) = Z. On the other hand, for ǫ ∈ 0,∞, ∂ǫ2(γ

τZ)

is a cycle obtained from γ1∂ǫ2(Z) by a permutation action. So, it is 0 because

∂ǫ2(Z) = 0 by the given assumptions. Similarly for j ≥ 4, we have ∂ǫj(γτZ) = 0.

Hence ∂(γτZ) = Z + τ · Z, as desired.Now let σ ∈ Sn be any. By a basic result from group theory, we can expressσ = τrτr−1 · · · τ2τ1, where each τi is a transposition of the form (p, p + 1)as considered before. Let σ0 := Id and σℓ := τℓτℓ−1 · · · τ1 for 1 ≤ ℓ ≤ r.For each such ℓ, by the previous case considered, we have (−1)ℓ−1σℓ−1 · Z +(−1)ℓ−1τℓ · σℓ−1 · Z = ∂((−1)ℓ−1γτℓσℓ−1·Z

). Since τℓ · σℓ−1 = σℓ, by taking the

sum of the above equations over all 1 ≤ ℓ ≤ r, after cancellations, we obtainZ + (−1)r−1σ · Z = ∂(γσZ), where γ

σZ :=

∑rℓ=1(−1)

ℓ−1γτℓσℓ−1·Z. Since (−1)r =

sgn(σ), we obtain the desired result.



5.2. Pontryagin product. Let X ∈ Schessk be an equidimensional scheme.

For m = (m1, · · · ,mr) ≥ 1, let CH(X [r]|Dm) := ⊕q,nCHq(X [r]|Dm, n). For

m ≥ 1, we let TCH(X ;m) = ⊕q,nTCHq(X,n;m) = ⊕q,nCH

q(X [1]|Dm+1, n−1). The objective of §5.2 is to prove the following result which generalizes [19,§3].

Theorem 5.4. Let k be a perfect field. Let m ≥ 0 and let m = (m1, · · · ,mr) ≥1. Let X,Y be both either in SmAff ess

k or in SmProjk. Then we have thefollowing:

(1) TCH(X ;m) is a graded commutative algebra with respect to a product∧X .

(2) CH(X [r]|Dm) is a graded module over TCH(X ; |m| − 1).(3) For f : Y → X with d = dimY − dimX, f∗ : CH(X [r]|Dm) →

CH(Y [r]|Dm) and f∗ : CH(Y [r]|Dm) → CH(X [r]|Dm)[−d] (if f isproper in addition) are morphisms of graded TCH(X ; |m|−1)-modules.

The proof requires a series of results and will be over after Lemma 5.13.

Lemma 5.5. Let X1, X2 ∈ Schessk . For i = 1, 2 and ri ≥ 1, let Vi be a cycle on

Xi×Ari×ni with modulus mi = (mi1, · · · ,miri), respectively. Then V1×V2,

regarded as a cycle on X1×X2×Ar1+r2 ×n1+n2 after a suitable exchange of

factors, has modulus (m1,m2).

Proof. We may assume that V1 and V2 are irreducible. It is enough to showthat each irreducible component W ⊂ V1 × V2 has modulus (m1,m2). Let

ιi : V i → Xi×Ari×ni

be the Zariski closure of Vi, and let νV i : VN

i → V i be

the normalization for i = 1, 2. Since k is perfect, [16, Lemma 3.1] says that the

morphism ν := νV 1×νV 2

: VN

1 ×VN

2 → V 1×V 2 = V1 × V2 is the normalization.

Hence, the compositeWN νW→ W

ι→ V 1×V 2, whereW is the Zariski closure of

W and νW is the normalization ofW , factors intoWN ιN→ V

N

1 ×VN

2ν→ V 1×V 2,

where ιN is the natural inclusion.Let (t1, · · · , tr1 , t

′1, · · · , t

′r2 , y1, · · · , yn1+n2

) ∈ Ar1+r2 × n1+n2

be the coor-

dinates. Consider two divisors D1 :=∑n1

i=1yi = 1 −∑r1j=1m1jtj =

0, D2 :=∑n1+n2

i=n1+1yi = 1 −∑r2j=1m2jt′j = 0. By the modulus con-

ditions satisfied by V1 and V2, we have ((ι1 × 1) (νV 1× 1))∗D1 ≥ 0 and

((1 × ι2) (1 × νV 2))∗D2 ≥ 0. Thus, we have ν∗(ι1 × ι2)∗(D1 + D2) ≥ 0 on

VN

1 ×VN

2 so that (ιN )∗ν∗(ι1×ι2)∗(D1+D2) ≥ 0 onW

N. Since ινW = ν ιN ,

this is equivalent to ν∗W ι∗(ι1× ι2)∗(D1+D2) ≥ 0, which showsW has modulus

(m1,m2).

Definition 5.6. Let r ≥ 1 be an integer and define µ : X1 × A1 × n1 ×

X2 × Ar ×n2 → X1 ×X2 × Ar ×

n1+n2 by (x1, t, yj)× (x2, ti, y′j) 7→(x1, x2, tti, yj, y′j).

The map µ is flat, but not proper. But, the following generalization of [19,Lemma 3.4] gives a way to take a push-forward:



Proposition 5.7. Let V1 ⊂ X1×A1×n1 and V2 ⊂ X2×Ar ×

n2 be closedsubschemes with moduli m and m ≥ 1, respectively. Then µ|V1×V2

is finite.

Proof. Since µ is an affine morphism, the proposition is equivalent to show thatµ|V1×V2

is projective.Set X = X1 ×X2 ×

n1+n2 . Let Γ → X1 ×X2 × A1 × Ar × Ar × n1+n2 =

X × A1 × Ar × Ar denote the graph of the morphism µ and let Γ → X ×P1 × (P1)r × (P1)r = X × P1 × P2 × P3 be its closure, where P1 = P1 andP2 = P3 = (P1)r. Let pi be the projection of X ×P1× (P1)r × (P1)r to X ×Pi

for 1 ≤ i ≤ 3. Set Γ0= p−1

3 (X × Ar). Then p3 : Γ0→ X × Ar is projective.

Using the homogeneous coordinates of P1 × P2 × P3, one checks easily that

Z := Γ0\Γ ⊂ E∪(

⋃ri=1 Ei) (the union is taken inside X×P1×P2×P3), where

E = X×∞×(0)r×Ar and Ei = X×0×((P1)i−1×∞×(P1)r−i)×Ar .Let V = V1×V2. Let ΓV be the graph Γ restricted to V and let ΓV be its Zariski

closure in X × P1 × P2 × P3. Since p3 : Γ0→ X × A

r is projective, so is the

map Γ0

V := ΓV ∩Γ0→ X×Ar. So, if we show Γ

0

V ∩Z = ∅, then V ≃ ΓV = Γ0

V

is projective over X × Ar, which is the assertion of the proposition.

To show Γ0

V ∩ Z = ∅, consider the projections X × P1 × P2 × P3p1→ X × P1

π1→X1×P1×

n1. Since the closure V 1 has modulus m ≥ 1 on X1×P1×n1, we

have V 1 ∩ (X1 × 0 × n1) = ∅. In particular, ΓV ∩ Ei → (π1 p1)−1(V 1 ∩

(X1 × 0 ×n1)) = ∅ for 1 ≤ i ≤ r.

To show that Γ0

V ∩ E = ∅, consider the projections X × P1 × P2 × P3p2→

X × P2π2→ X2 × P2 ×

n2 . Since the closure V 2 has modulus m ≥ 1 onX2×P2×

n2, we have V 2∩ (X2× (0)r×n2) = ∅. In particular, ΓV ∩E →

(π2 p2)−1(V 2 ∩ (X2 × (0)r ×n2)) = ∅. This finishes the proof.

Lemma 5.8. Let X ∈ Schessk and let V be a cycle on X × A1 × Ar ×

n withmodulus (|m|,m), where m = (m1, · · · ,mr) ≥ 1. Suppose µ|V is finite. Thenthe closed subscheme µ(V ) on X × Ar ×

n has modulus m.

Proof. This is a straightforward generalization of [19, Proposition 3.8] and is asimple application of Lemma 2.7. We skip the detail. We only remark that it iscrucial for the proof that the A1-component of the modulus is at least |m|.

Definition 5.9. For any irreducible closed subscheme V ⊂ X ×A1×Ar×n

such that µ|V : V → µ(V ) is finite, where µ is as in Definition 5.6, define µ∗(V )as the push-forward µ∗(V ) = deg(µ|V ) · [µ(V )]. Extend it Z-linearly.If V1 is a cycle on X1×A1×

n1 and V2 is a cycle on X2×Ar×n2 such that

µ|V1×V2is finite, we define the external product V1 ×µ V2 := µ∗(V1 × V2). If

pi = dim Vi, then dim(V1 ×µ V2) = p1 + p2. If X1 ×X2 is equidimensional andif qi is the codimension of Vi, then V1 ×µ V2 has codimension q1 + q2 − 1.

Lemma 5.10. Let V1 ∈ zq1(X1[1]|Dm, n1) and V2 ∈ zq2(X2[r]|Dm, n2) withX1, X2 ∈ Schess

k and m,m ≥ 1. Then V1 ×µ V2 intersects all faces of X1 ×X2 × Ar ×

n1+n2 properly.



Proof. We may assume that V1 and V2 are irreducible. V1×V2 clearly intersectsall faces of X1×X2×A1×Ar×n1+n2 properly. It follows from Proposition 5.7that µ|V1×V2

is finite. In this case, the proper intersection property of µ(V1 ×µV2) follows exactly like that of the finite push-forwards of Bloch’s higher Chowcycles.

Corollary 5.11. Let X1, X2, X3 ∈ Schessk be equidimensional and let m ≥ 1.

Then there is a product

×µ : zq1(X1[1]|D|m|, n1)⊗ zq2(X2[r]|Dm, n2)→

→ zq1+q2−1((X1 ×X2)[r]|Dm, n1 + n2)

which satisfies the relation ∂(ξ ×µ η) = ∂(ξ) ×µ η + (−1)n1ξ ×µ ∂(η). Itis associative in the sense that (α1 ×µ α2) ×µ β = α1 ×µ (α2 ×µ β) forαi ∈ zqi(Xi[1]|D|m|, ni) for i = 1, 2 and β ∈ zq3(X3[r]|Dm, n3). In particu-lar, it induces operations ×µ : CHq1(X1[1]|D|m|, n1) ⊗ CHq2(X2[r]|Dm, n2) →

CHq1+q2−1((X1 ×X2)[r]|Dm, n1 + n2).

Proof. The existence of ×µ on the level of cycle complexes follows from thecombination of Proposition 5.7, Lemma 5.8 and Lemma 5.10. The associativityfollows from that of the Cartesian product × and the product µ : A1×A1 → A1.By definition, one checks ∂(ξ×η) = ∂(ξ)×η+(−1)n1ξ×∂(η). So, by applyingµ∗, we get the required relation. That ×µ descends to the homology follows.

Definition 5.12. Let m = (m1, · · · ,mr) ≥ 1 and let X be in SmAff essk

or in SmProjk. For cycle classes α1 ∈ CHq1(X [1]|D|m|, n1) and α2 ∈CHq2(X [r]|Dm, n2), define the internal product α1 ∧X α2 to be ∆∗

X(α1 ×µ α2)

via the diagonal pull-back ∆∗X : CHq1+q2−1((X × X)[r]|Dm, n1 + n2) →

CHq1+q2−1(X [r]|Dm, n1 + n2). This map exists by Theorem 4.5 and Corol-lary 4.15.

Lemma 5.13. ∧X is associative in the sense that (α1 ∧X α2) ∧X β = α1 ∧X(α2 ∧X β) for α1, α2 ∈ CH(X [1]|D|m|) and β ∈ CH(X [r]|Dm). ∧X is alsograded-commutative on CH(X [1]|D|m|).

Proof. The associativity holds by Corollary 5.11. For the graded-commutativity, first note by Theorem 3.2 that we can find representativesα1 and α2 of the given cycle classes whose codimension 1 faces are all triv-ial. Let σ be the permutation that sends (1, · · · , n1, n1 + 1, · · · , n1 + n2) to(n1 + 1, · · · , n1 + n2, 1, · · · , n1) so that sgn(σ) = (−1)n1+n2 . It follows fromLemma 5.3 that α1 ∧X α2 = (−1)n1+n2α2 ∧X α1 + ∂(W ) for some admissiblecycle W , as desired.

Proof of Theorem 5.4. The proof of (1) and (2) is just a combinationof the above discussion under the observation that TCHq(X,n;m) =CHq(X [1]|Dm+1, n− 1) for m ≥ 0 and n ≥ 1. To prove (3) for f∗, consider thecommutative diagram



(5.3) Y [r]×n

f

∆Y // (Y × Y )[r]×n

f×f

(Y × Y )[r + 1]×n

f×f

µYoo

X [r]×n ∆X // (X ×X)[r]×

n (X ×X)[r + 1]×n.

µXoo

There is a finite set W of locally closed subsets of X such thatf∗ : zq1W(X [1]|D|m|, •) → zq1(Y [1]|D|m|, •) and f∗ : zq2W(X [r]|Dm, •) →zq2(Y [r]|Dm, •) can be defined as taking cycles associated to the inverse images.Moreover, it is enough to consider the product of cycles in zq1W(X [1]|D|m|, •)and zq2W(X [r]|Dm, •) by the moving lemmas Theorems 4.1 and 4.4. For irre-ducible cycles V1 ∈ zq1(X [1]|D|m|, n1) and V2 ∈ zq2(X [r]|Dm, n2), the mapµY is finite when restricted to f∗(V1) × f

∗(V2) by Lemma 5.7. In particular,µY (f

∗(V1)× f∗(V2)) ∈ zq1+q2−1((Y × Y )[r]|Dm, n1 + n2).Since the right square in the diagram (5.3) is transverse, it follows thatf∗(µX(V2 × V2)) = µY (f

∗(V1) × f∗(V2)) as cycles. The desired commuta-tivity of the product with f∗ now follows from the commutativity of the leftsquare in (5.3) and the composition law of Theorem 4.5.The proof of (3) for f∗ is just the projection formula, whose proof is identical tothe one given in [19, Theorem 3.19] in the case when X1, X2 ∈ SmProjk.

As applications, we obtain:

Corollary 5.14. Let X be in SmAffessk or in SmProjk. Then for q, n ≥ 0

and m ≥ 1, the group CHq(X [r]|Dm, n) is a W(|m|−1)(k)-module.

Proof. Applying Theorem 5.4 to X and the structure map X → Spec (k),it follows that CH(X [r]|Dm) is a graded module over TCH(k; |m| − 1). By

Corollary 5.11, this yields a TCH1(k, 1; |m| − 1)-module structure on eachCHq(X [r]|Dm, n). The corollary now follows from the fact that there is a

ring isomorphism Wm(k)∼→ TCH1(k, 1;m) for every m ≥ 1 by [28, Corol-

lary 3.7].

We can explain the homotopy invariance of the groups CHq(X,n) in terms ofadditive higher Chow groups as follows.

Corollary 5.15. For X ∈ Schessk which is equidimensional and for q, n ≥ 0,

we have CHq(X [1]|D1, n) = 0.

Proof. By Corollary 5.11, we have a map ×µ : CH1(pt[1]|D1, 0) ⊗CHq(X [1]|D1, n) → CHq(X [1]|D1, n) and it follows from the definition of ×µthat [1]×µ α = α for every α ∈ CHq(X [1]|D1, n), where [1] ∈ CH1(pt[1]|D1, 0)is the cycle given by the closed point 1 ∈ A1(k). It therefore suffices to showthat the homology class of 1 is zero. To do so, we may use the identification(, ∞, 0) ≃ (A1, 0, 1) given by y 7→ 1/(1 − y) again. Then the cycleC ⊂ A2 given by (t, y) ∈ A2|ty = 1 is an admissible cycle in z1(pt[1]|D1, 1)such that ∂1([C]) = [1] and ∂0([C]) = 0.



6. The structure of differential graded modules

In this section, we construct a differential operator on the graded module of§5 of multivariate additive higher Chow groups over the univariate additivehigher Chow groups, generalizing [19, §4]. We assume that k is perfect andchar(k) 6= 2.

6.1. Differential. Let X be a smooth quasi-projective scheme essentially offinite type over k. Let r ≥ 1 and let m = (m1, · · · ,mr) ≥ 1. Let (Grm)× :=(t1, · · · , tr) ∈ Grm | t1 · · · tr 6= 1. Consider the morphism δn : (Grm)××

n →Grm ×

n+1, (t1, · · · , tr, y1, · · · , yn) 7→ (t1, · · · , tr,1

t1···tr, y1, · · · , yn). It induces

δn : X × (Grm)× ×n → X ×Grm ×

n+1.Recall a closed subscheme Z ⊂ X×Ar×

n with modulus m does not intersectthe divisor t1 · · · tr = 0. So, it is closed in X × Grm ×

n. For such Z, wedefine Z× := Z|X×(Grm)××n .

Lemma 6.1. For a closed subscheme Z ⊂ X × Ar × n with modulus m, the

image δn(Z×) is closed in X ×Grm ×

n+1.

Proof. It is enough to show that δn : X × (Grm)××n → X ×Grm×

n+1 is aclosed immersion. It reduces to show that the map (Grm)× → Grm × (P1 \ 1)given by (t1, · · · , tr) 7→ (t1, · · · , tr, 1/(t1 · · · tr)) is a closed immersion. This isobvious because the image coincides with the closed subscheme given by theequation t1 · · · try = 1, where (t1, · · · , tr, y) ∈ G

rm× are the coordinates.

Definition 6.2 (cf. [19, Definition 4.3]). For a closed subscheme Z ⊂ X ×Ar×

n with modulus m, we write δn(Z) := δn(Z×). If Z is a cycle, we define

δn(Z) by extending it Z-linearly. We may often write δ(Z) if no confusionarises.

Lemma 6.3. Let Z be a cycle on X × Ar × n with modulus m. Then δn(Z)

is a cycle on X × Ar ×n+1 with modulus m.

Proof. We may suppose that Z is irreducible. Let V = δn(Z), which is a prioriclosed in X×G

rm×

n+1. If the closure V ′ of V in X×Ar×

n+1 has modulusm, then it does not intersect the divisor t1 · · · tr = 0 of X × Ar ×

n+1, soV = V ′, and V is closed in X ×Ar ×

n+1 with modulus m. So, we reduce toshow that V ′ has modulus m.Let Z and V be the Zariski closures of Z and V ′ in X × Ar ×

nand X ×

Ar × n+1

, respectively. Observe that δn extends to δn : X × Ar × n→

X × Ar × n+1

, which is induced from ArΓ→ Ar ×

Id×σ→ Ar × , where

Γ is the graph morphism of the composite Ar→A1 → of the product mapfollowed by the open inclusion, (t1, · · · , tr) 7→ (t1 · · · tr) 7→ (t1 · · · tr; 1), whileσ : → is the antipodal automorphism (a; b) 7→ (b; a), where (a; b) ∈ = P1

are the homogeneous coordinates. Since Γ is a closed immersion and Id × σis an isomorphism, the morphism δn is projective. Hence, the dominant mapδn|Z× : Z× → V induces δn|Z : Z → V . In particular, we have a commutative



diagram

(6.1) ZN

δn

νZ // Z

δn|Z

ιZ // X × Ar ×n

δn

VN νV // V

ιV // X × Ar ×n+1

,

where ιZ , ιV are the closed immersions, νZ , νV are normalizations, and δn isgiven by the universal property of normalization for dominant maps.

By definition, δ∗

ntj = 0 = tj = 0 for 1 ≤ j ≤ r. First con-

sider the case n ≥ 1. Then δ∗

nF1n+1,i = F 1

n,i−1 for 2 ≤ i ≤ n + 1.

Now, δ∗nν∗V ι

∗V (

∑n+1i=1 F

1n+1,i −

∑rj=1mjtj = 0) ≥ δ∗nν

∗V ι

∗V (

∑n+1i=2 F

1n+1,i −∑r

j=1mjtj = 0) =† ν∗Zι∗Zδ

∗

n(∑n+1

i=2 F1n+1,i −

∑rj=1mjtj = 0) =

ν∗Zι∗Z(

∑n+1i=2 F

1n,i−1 −

∑rj=1mjtj = 0) = ν∗Zι

∗Z(

∑ni=1 F

1n,i −

∑rj=1mjtj =

0) ≥‡ 0, where † holds by the commutativity of (6.1) and ‡ holds as Z has

modulus m. Using Lemma 2.7, we can drop δ∗n, i.e., V′ has modulus m.

When n = 0, we have for 1 ≤ j ≤ r, δ∗0ν∗V ι

∗V tj = 0 = ν∗Zι

∗Zδ

∗

0tj = 0 =

ν∗Zι∗Ztj = 0, which is 0 because Z ∩ tj = 0 = ∅. Hence, δ∗0ν

∗V ι

∗V (F

11,1 −∑r

j=1mjtj = 0) = δ∗0ν∗V ι

∗V F

11,1 ≥ 0. Dropping δ∗0 , we get V ′ has modulus

m.

Proposition 6.4. Let Z ∈ zq(X [r]|Dm, n). Then δ(Z) ∈ zq+1(X [r]|Dm, n+1).Furthermore, δ and ∂ satisfy the equality δ∂ + ∂δ = 0.

Proof. We may assume that Z is an irreducible cycle. Let ∂ǫn,i be the boundarygiven by the face F ǫn,i on X × Ar ×

n, for 1 ≤ i ≤ n and ǫ = 0,∞.Claim: For ǫ = 0,∞, (i) ∂ǫn+1,1 δn = 0, (ii) ∂ǫn+1,i δn = δn−1 ∂ǫn,i−1 for2 ≤ i ≤ n+ 1.For (i), we show that δn(Z) ∩ y1 = ǫ = ∅ for ǫ = 0,∞. Since δn(Z) ⊂V (t1 · · · try1 = 1), we have δn(Z) ∩ y1 = 0 = ∅. On the other hand, if δn(Z)intersects y1 =∞, then some ti must be zero on Z, i.e., Z intersects ti = 0for some 1 ≤ i ≤ r. However, since Z has modulus m, this can not happen.Thus, δn(Z) ∩ y1 =∞ = ∅. This shows (i). For (ii), by the definition of δn,the diagram

(Grm)× ×n−1

ιǫi−1//

δn−1

(Grm)× ×

n

δn

Grm ×n

ιǫi // Grm ×n+1

is Cartesian. Thus, δn−1((ι∗i−1(Z)) = (ιǫi)

∗(δn(Z)) by [6, Proposition 1.7], i.e.,(ii) holds. This proves the claim.By Lemma 6.3, we know δn(Z) has modulus m. Since Z intersects all facesproperly, so does δn(Z) by applying (i) and (ii) of the above claim repeatedly.

For ∂δ+δ∂ = 0, note that ∂δn(Z) =∑n+1

i=1 (−1)i(∂∞n+1,iδn(Z)−∂

0n+1,iδn(Z)) =

†



∑n+1i=2 (−1)

i(δn−1∂∞n,i−1(Z) − δn−1∂

0n,i−1(Z)) = −

∑ni=1(−1)

i(δn−1∂∞n,i(Z) −

δn−1∂0n,i−1(Z)) = −δn−1

∑ni=1(−1)

i(∂∞n,i(Z)− ∂0n,i(Z)) = −δn−1 ∂(Z), where

† holds by the claim.

Lemma 6.5 and Corollary 6.6 below, which generalize [19, §4.2], have muchsimpler proofs than loc.cit.

Lemma 6.5. Let Z ∈ zq(X [r]|Dm, n) be such that ∂ǫi (Z) = 0 for 1 ≤ i ≤ n andǫ = 0,∞. Then 2δ2(Z) is the boundary of an admissible cycle with modulus m.

Proof. Note that δ2(Z) is an admissible cycle on X×Ar×n+2 with modulus

m, by Proposition 6.4. For the transposition τ = (1, 2) on the set 1, · · · , n+2,we have τ · δ2(Z) = δ2(Z), by the definition of δ. On the other hand, we haveτ · δ2(Z) = −δ2(Z) + ∂(γ) for some admissible cycle γ, by Lemma 5.3. Hence,we have −δ2(Z) + ∂(γ) = δ2(Z), i.e., 2δ2(Z) = ∂(γ), as desired.

Corollary 6.6. Let k be a perfect field of characteristic 6= 2 and let X be inSmAff ess

k or in SmProjk. Let m ≥ 1. Then δ2 = 0 on CHq(X [r]|Dm, n).

Proof. If r = m = 1, by Corollary 5.15, there is nothing to prove. So, supposeeither r ≥ 2 or |m| ≥ 2. But, if r ≥ 2, then we automatically have |m| ≥ 2, sowe just consider the latter case.Given α ∈ CHq(X [r]|Dm, n), by Theorem 3.2, we can find a representativeZ ∈ zq(X [r]|Dm, n) such that ∂ǫi (Z) = 0 for 1 ≤ i ≤ n and ǫ = 0,∞. Then byLemma 6.5, we have 2δ2(α) = 0.On the other hand, by Corollary 5.14, the group CHq(X [r]|Dm, n) is aW(|m|−1)(k)-module. As |m| ≥ 2 and char(k) 6= 2, it follows that 2 ∈(W(|m|−1)(k))

×. In particular, δ2(α) = 0.

6.2. Leibniz rule. We now discuss the Leibniz rule, generalizing [19, §4.3].Let X ∈ Schess

k . Let (x, t, t1, · · · , tr, y1, · · · , yn+2) ∈ X × Ar+1 × n+2 be the

coordinates. Let T ⊂ X × Ar+1 × n+2 be the closed subscheme defined by

the equation tyn+1 = yn+2(tt1 · · · tryn+1 − 1).

Definition 6.7 (cf. [19, Definition 4.9]). Given a closed subscheme Z ⊂X × Ar+1 ×

n, define CZ := T · (Z × 2) on X × Ar+1 ×

n+2. This isextended Z-linearly to cycles.

Lemma 6.8. Let Z be a cycle on X × Ar+1 × n with modulus m =

(m1, · · · ,mr+1). Then CZ has modulus m on X × Ar+1 ×n+2.

Proof. We may assume Z is irreducible. We show that each irreducible com-ponent V ⊂ CZ has modulus m. Let Z and V be the Zariski closures of Z

and V in X × Ar+1 ×

nand X × A

r+1 ×n+2

, respectively. The projection

pr : X × Ar+1 ×

n+2→ X × A

r+1 × nthat ignores the last two

2is pro-

jective, while its restriction to X × Ar+1 ×n+2 maps V into Z. So, pr maps



V to Z, giving a commutative diagram

(6.2) VN νV

//

prN

V ιV

//

pr|V

X × Ar+1 ×n+2

pr

ZN νZ

// Z ιZ

// X × Ar+1 ×n,

where ιV and ιZ are the closed immersions, νV and νZ are normalizations, andprN is induced by the universal property of normalization for dominant maps.The modulus condition for V is now easily verified using the pull-back of the

modulus condition for Z on ZN

and the fact that pr∗tj = 0 = tj = 0 forall j and pr∗F 1

n,i = F 1n+2,i for all i.

Corollary 6.9. Let X1, X2 ∈ Schessk . Let V1 ⊂ X1 × A1 ×

n1 and V2 ⊂X2 × Ar ×

n2 be closed subschemes with moduli |m| and m, respectively withm ≥ 1.Under the exchange of factors X1×A1×n1×X2×Ar×n2 ≃ X1×X2×Ar+1×n, where n = n1 +n2, consider the cycle CV1×V2

on X1×X2×Ar+1×n+2.

Then µ|CV1×V2is finite. In particular, µ∗(CV1×V2

) as in Definition 5.9 is well-defined, and has modulus m.

Proof. We set V = V1 × V2. From the definition of µ, the map µ : V × 2 →

X1 ×X2 × Ar ×

n+2 is of the form µ|V × Id2 . By Proposition 5.7, the mapµ|V is finite, thus so is µ|V × Id2 : V ×

2 → X1 ×X2 × Ar × n+2. Hence,

its restriction to CV = T · (V × 2) is also finite. The modulus condition for

µ∗(CV ) follows from Lemmas 5.8 and 6.8.

Definition 6.10 (cf. [19, Definition 4.12]). Let V1 ∈ zq1(X1[1]|D|m|, n1) andV2 ∈ zq2(X2[r]|Dm, n2) with X1, X2 ∈ Schess

k . Let n = n1 + n2 and defineV1×µ′ V2 be the cycle σ ·µ∗(CV1×V2

), where σ = (n+2, n+1, · · · , 1)2 ∈ Sn+2.

Lemma 6.11. Let V1, V2 be as in Definition 6.10. Then V1 ×µ′ V2 ∈zq1+q2−1((X1 ×X2)[r]|Dm, n1 + n2 + 2).

Proof. By Corollary 6.9, the cycle µ∗(CV1×V2) has modulus m, thus so does

W := V1 ×µ′ V2. It remains to prove that W intersects all faces properly. Letσn1

= (n1 + 1, n1, · · · , 1) ∈ Sn+1. Then by direct calculations, we have(6.3)

∂∞1 W = σn1(V1 ×µ δ(V2)), ∂01W = 0, ∂∞2 W = δ(V1 ×µ V2),

∂02W = δ(V1)×µ V2,

∂ǫiW =

∂ǫi−2(V1)×µ′ V2, for 3 ≤ i ≤ n1 + 2,V1 ×µ′ ∂ǫi−n1−2(V2), for n1 + 3 ≤ i ≤ n+ 2,

ǫ ∈ 0,∞.

Since each Vi is admissible, using (6.3), Lemma 5.10, Proposition 6.4 and in-duction on the codimension of faces, we deduce that W intersects all facesproperly.

Proposition 6.12. Let X1, X2 ∈ Smessk . Let ξ ∈ zq1(X1[1]|D|m|, n1) and

η ∈ zq2(X2[r]|Dm, n2). Let n = n1 + n2 and q = q1 + q2. Suppose that



all codimension one faces of ξ and η vanish. Then in the group zq−1((X1 ×X2)[r]|Dm, n+1), the cycle δ(ξ×µ η)−δξ×µ η− (−1)n1ξ×µ δη is the boundaryof an admissible cycle.

Proof. By (6.3), for 3 ≤ i ≤ n1 + 2, we have ∂ǫi (ξ ×µ′ η) = ∂ǫi−2(ξ) ×µ′ η = 0,while for n1+3 ≤ i ≤ n+2, we have ∂ǫi (ξ×µ′ η) = ξ×µ′ ∂ǫi−n1−2(η) = 0. Hence,

∂(ξ×µ′ η) =∑n+2

i=1 (−1)i(∂∞i −∂

0i )(ξ×µ′ η) = δ(ξ×µη)−σn1

·(ξ×µδη)+δξ×µηby (6.3) for i = 1, 2. Equivalently,

(6.4) δ(ξ ×µ η)− δξ ×µ η − σn1· (ξ ×µ δη) = ∂(ξ ×µ′ η).

But, for ξ ×µ δη, notice that

(6.5) ∂ǫi (ξ ×µ δη) =

∂ǫi ξ ×µ δη = 0, for 1 ≤ i ≤ n1,ξ ×µ ∂ǫi−n1

(δη), for n1 + 1 ≤ i ≤ n+ 1,ǫ ∈ 0,∞.

We have ∂ǫ1(δη) = 0 when i = n1 + 1 by Claim (i) of Proposition 6.4, and∂ǫi−n1

(δη) = δ(∂ǫi−n1−1η) = δ(0) = 0 when n1 + 2 ≤ i ≤ n + 1 by Claim(ii) of Proposition 6.4. Hence, ξ ×µ δη is a cycle with trivial codimension 1faces, so, by Lemma 5.3, for some admissible cycle γ, we have σn1

· (ξ×µ δη) =sgn(σn1

)(ξ×µ δη)+∂(γ) = (−1)n1ξ×µ δη+∂(γ). Putting this back in (6.4), weobtain δ(ξ×µ η)− δξ×µ η− (−1)n1ξ×µ δη = ∂(ξ×µ′ η)− ∂(γ), as desired.

The above discussion summarizes as follows:

Theorem 6.13. Let X be in SmAff essk or in SmProjk over a perfect field k

with char(k) 6= 2. Let r ≥ 1 and m = (m1, · · · ,mr) ≥ 1. Then the followinghold:

(1) (CH(X [1]|D|m|),∧X , δ) forms a commutative differential gradedW(|m|−1)Ω

•k-algebra.

(2) (CH(X [r]|Dm), δ) forms a differential graded (CH(X [1]|D|m|),∧X , δ)-module.

In particular, (CH(X [r]|Dm), δ) is a differential graded W(|m|−1)Ω•k-module.

Proof. The commutative differential graded algebra structure onCH(X [1]|D|m|) and the differential graded module structure on CH(X [r]|Dm)over CH(X [1]|D|m|) follows by combining Theorem 5.4, Corollary 6.6 andProposition 6.12 using Theorem 3.2.The structure map p : X → Spec (k) turns (CH(X [1]|D|m|),∧X , δ)into a differential graded algebra over (CH(pt[1]|D|m|),∧pt, δ) via p∗.

Since ⊕n≥0CHn+1(pt[1]|D|m|, n) forms a differential graded sub-algebra of

(CH(pt[1]|D|m|),∧pt, δ). The map of commutative differential graded algebras

W(|m|−1)Ω•k → ⊕n≥0CH

n+1(pt[1]|D|m|, n) (see [28]) finishes the proof of thetheorem.

As a consequence of Theorem 6.13 (use Corollary 5.15 when |m| = 1), we obtainthe following property of multivariate additive higher Chow groups.



Corollary 6.14. Let r ≥ 1 and m ≥ 1 and let X be in SmAffessk or in

SmProjk. Then each CHq(X [r]|Dm, n) is a k-vector space provided char(k) =0.

7. Witt-complex structure on additive higher Chow groups

Let k be a perfect field of characteristic 6= 2. In this section, a smooth affinek-scheme means an object in SmAffess

k , i.e., an object of either SmAffk orSmLock.Rulling proved in [28] that the additive higher Chow groups of 0-cycles overSpec (k) form a restricted Witt-complex over k. When X is a smooth projectivevariety over k, it was proven in [19] that additive higher Chow groups of Xform a restricted Witt-complex over k. Our objective is to prove the strongerassertion that the additive higher Chow groups of Spec (R) ∈ SmAffess

k havethe structure of a restricted Witt-complex over R.Since we exclusively use the case r = 1 only, we use the older no-tations Tzq(X,n;m) and TCHq(X,n;m) instead of zq(X [1]|Dm+1, n − 1)and CHq(X [1]|Dm+1, n − 1). For X ∈ Schess

k , we let TCH(X ;m) :=

⊕n,qTCHq(X,n;m) and TCHM (X ;m) := ⊕nTCH

n(X,n;m). The super-

script M is for Milnor. Let TCH(X) := ⊕mTCH(X ;m) and TCHM (X) :=

⊕mTCHM (X ;m). We similarly define T CH(X ;m), T CHM (X ;m), T CH(X),

and T CHM (X) for X ∈ Schk using Definition 4.7.

7.1. Witt-complex structure over k. In this section, we show that theadditive higher Chow groups for an object of SmAffess

k form a functorial re-stricted Witt-complex over k. For r ≥ 1, let φr : A

1 → A1 be the morphism x 7→

xr, which induces φr : Spec (R)× Bn → Spec (R)× Bn. By [19, §5.1, 5.2], wehave the Frobenius Fr : TCH

q(R, n; rm+r−1)→ TCHq(R, n;m) and the Ver-schiebung Vr : TCH

q(R, n;m)→ TCHq(R, n; rm+r−1) given by Fr = φr∗ andVr = φ∗r . We also have a natural inclusion R : Tzq(R, •;m+1)→ Tzq(R, •;m)for any m ≥ 1, which induces R : TCHq(R, n;m + 1) → TCHq(R, n;m),called the restriction. Finally, by Theorem 6.13, there is a differentialδ : Tzq(R, •;m) → Tzq(R, • + 1;m), which induces δ : TCHq(R, n;m) →TCHq(R, n+ 1;m).

Theorem 7.1. Let X ∈ SmAffessk and m ≥ 1. Then TCH(X ;m) is a DGA

and TCHM (X ;m) is its sub-DGA. Furthermore, with respect to the operationsδ,R, Fr, Vr in the above together with λ = f∗ : Wm(k) = TCH1(k, 1;m) →TCH1(X, 1;m) for the structure morphism f : X → Spec (k), TCH(X) is a

restricted Witt-complex over k and TCHM (X) is a restricted sub-Witt-complexover k. These structures are functorial.

Proof. In [19, Theorem 1.1, Scholium 1.2], it was stated that TCH(X ;m) and

TCHM (X ;m) are DGAs, and that TCH(X) and TCHM (X) are restrictedWitt-complexes over k with respect to the above δ,R, Fr, Vr, provided themoving lemma holds for X . But this is now shown in Theorems 4.1 and 4.10.We give a very brief sketch of this structure and its functoriality.



The functoriality of the restriction operator R recalled above, was stated in[19, Corollary 5.19], which we easily check here: let f : X → Y be a morphismin SmAff ess

k and consider the following commutative diagram:

TzqW(Y, •;m+ 1)f∗

// _

Tzq(X, •;m+ 1) _

TzqW(Y, •;m)f∗

// Tzq(X, •;m),

where W is a finite set of locally closed subsets of Y , and the horizontal mapsare chain maps given by the inverse images as in the proof of Theorem 4.5 andCorollary 4.15. The diagram and Theorems 4.1 and 4.10 imply that f∗R = Rf∗

because the vertical inclusions induce R by definition.For each r ≥ 1, the Frobenius Fr and Verschiebung Vr recalled in the aboveare functorial as proven in [19, Lemmas 5.4, 5.9], and that Fr is a graded ringhomomorphism is proven in [19, Corollary 5.6].Finally, the properties (i), (ii), (iii), (iv), (v) in Section 2.2.2, are all proven in[19, Theorem 5.13], where none requires the projectivity assumption.

Corollary 7.2. Let m ≥ 1 be an integer. Then T CH(−;m) and

T CHM (−;m) define presheaves of DGAs on Schk, and the pro-systems

T CH(−) and T CHM (−) define presheaves of restricted Witt-complexes overk on Schk.

Proof. Let X ∈ Schk. By definition, T CH(X ;m) is the colimit over all(X → A) ∈ (X ↓ SmAffk)

op of TCH(A;m). But the category of DGAs isclosed under filtered colimits (see [13]) so that T CH(X ;m) is a DGA. For eachmorphism f : X → Y in Schk, one checks f

∗ : T CH(Y ;m)→ T CH(X ;m) is amorphism of DGAs. The other assertions follow easily using Theorem 7.1.

Before we discuss Witt-complexes over R, we state the following behavior ofvarious operators under finite push-forward maps.

Proposition 7.3. Let f : X → Y be a finite map in SmAffessk . Then for r ≥ 1,

we have: (a) f∗R = Rf∗; (b) f∗δ = δf∗; (c) f∗Fr = Frf∗; (d) f∗Vr = Vrf∗.

Proof. The item (a) is obvious and (b) and (c) follow at once from the factthat these operators are defined as push-forward under closed immersion andfinite maps and they preserve the faces. For (d), we consider the commutativediagram

(7.1) X × A1

Id×φr//

f×Id

X × A1

f×Id

Y × A1Id×φr

// Y × A1.

Since this diagram is Cartesian and f as well as φ preserve the faces, we con-clude from [6, Proposition 1.7] that f∗ φ∗r = φ∗r f∗.



7.2. Witt-complex structure over R. Let X = Spec (R) ∈ SmAffessk .

The objective of this section is to strengthen Theorem 7.1 by showing thatTCH(X) is a restricted Witt-complex over R.

Letm ≥ 1 be an integer. We first define a group homomorphism τR : Wm(R)→TCH1(R, 1;m) for any k-algebraR. Recall that the underlying abelian group ofWm(R) identifies with the multiplicative group (1+tR[[t]])×/(1+tm+1R[[t]])×.For each polynomial p(t) ∈ (1 + R[[t]])×, consider the closed subscheme ofSpec (R[t]) given by the ideal (p(t)), and let Γ(p(t)) be its associated cycle.

By definition, Γ(p(t)) ∩ t = 0 = ∅ so that Γ(p(t)) ∈ Tz1(R, 1;m). We setΓa,n = Γ(1−atn) for n ≥ 1 and a ∈ R.

Lemma 7.4. Let f(t), g(t) be polynomials in R[t], and let h(t) ∈ R[t] be theunique polynomial such that (1−tf(t))(1−tg(t)) = 1−th(t). Then Γ(1−th(t)) =

Γ(1−tf(t)) + Γ(1−tg(t)) in Tz1(R, 1;m).

Proof. This is obvious by (1− tf(t))(1 − tg(t)) = 1− th(t).

Lemma 7.5. For n ≥ m+ 1, we have Γ(1−tnf(t)) ≡ 0 in TCH1(R, 1;m).

Proof. Consider the closed subscheme Γ ⊂ X×A1× given by y1 = 1−tnf(t).

Let ν : ΓN→ Γ → X×A1×P1 be the normalization of the Zariski closure Γ in

X ×A1×P1. Since f(t)tn = 1− y1 on Γ, we see that nν∗t = 0 ≤ ν∗y1 = 1

on ΓN. Since n ≥ m+ 1, this shows that Γ satisfies the modulus m condition.

Since ∂∞1 (Γ) = 0 and ∂01(Γ) = Γ(1−tnf(t)) (which is of codimension 1), the cycle

Γ is an admissible cycle in Tz1(R, 2;m) such that ∂Γ = Γ(1−tnf(t)). This shows

that Γ(1−tnf(t)) ≡ 0 in TCH1(R, 1;m).

Proposition 7.6. Let R be a k-algebra. Then the map τR : (1 + tR[t]) →Tz1(R, 1;m) that sends a polynomial 1 − tf(t) to Γ(1−tf(t)), defines a group

homomorphism τR : Wm(R)→ TCH1(R, 1;m).

Proof. Every element p(t) ∈ (1 + tR[[t]])× has a unique expression p(t) =∏n≥1(1− ant

n) for an ∈ R. For any such p(t), set p≤m(t) =∏mn=1(1 − ant

n).

We define τR(p(t)) = Γ(p≤m(t)). It follows from Lemmas 7.4 and 7.5 that thismap descends to a group homomorphism from Wm(R).

Recall from [28, Appendix A] that for each r ≥ 1, we have the Frobenius Fr :Wrm+r−1(R) → Wm(R) and the Verschiebung Vr : Wm(R) → Wrm+r−1(R).They are given by Fr(1 − atn) = (1− a

rs t

ns )s, where s = gcd(r, n) and Vr(1−

atn) = 1− atrn. On the other hand, as seen in Section 7.1, we have operationsFr and Vr on TCH1(R, 1;m)m∈N.

Lemma 7.7. Let R be a k-algebra. Then the maps τR : Wm(R) →TCH1(R, 1;m) of Proposition 7.6 commute with the Fr and Vr operators onboth sides.



Proof. That τRVr = VrτR, is easy: we have Vr(τR(1−atn)) = Vr(Γa,n) = Γa,rn,while τR(Vr(1− atn)) = Γ(1−atrn) = Γa,rn.That τRFr = FrτR, is slightly more involved. Recall that Fr(1 − atn) =(1−a

rs t

ns )s, where s = gcd(r, n). Write n = n′s and r = r′s, where 1 = (r′, n′).

Hence, we have τRFr(1 − atn) = sΓars ,ns

= sVns(Γars ,1

) = sVn′(Γar′ ,1) =: ♣,

while FrτR(1− atn) = FrΓa,n = FrVn(Γa,1) =: ♥.First observe that when n = 1, we have s = 1, r = r′, n = n′ = 1, and we have♥ = Fr(Γa,1) = Γar,1 = ♣, so that τRFr(1− at) = FrτR(1− at), indeed.For a general n ≥ 1, we have FrVn = Fr′FsVsVn′ = Fr′ (s · Id) Vn′ =sFr′Vn′ =† sVn′Fr′ , where † holds because (r′, n′) = 1. Since Fr′(Γa,1) = Γar′ ,1(by the first case), we have ♥ = FrVn(Γa,1) = sVn′Fr′(Γa,1) = sVn′(Γar′ ,1) =♣. This shows τRFr = FrτR.

Remark 7.8. In the proof of Lemma 7.7, we saw that for s = (r, n),

(7.2) Fr(Γa,n) = sΓars ,ns

, Vr(Γa,n) = Γa,rn.

Proposition 7.9. For X = Spec (R) ∈ SmAffessk , the maps τR : Wm(R) →

TCH1(R, 1;m) form a morphism of pro-rings that commutes with Fr and Vrfor r ≥ 1.

Proof. It is clear from the definition of τR in Proposition 7.6 that it commuteswith R. We saw that τR commutes with Fr and Vr in Lemma 7.7. So, weonly need to show that τR respects the products. By [2, Proposition (1.1)], itis enough to prove that for a, b ∈ R and u, v ≥ 1,

(7.3) Γa,u ∧ Γb,v = wΓavw b

uw ,uvw

in TCH1(R, 1;m),

where w = gcd(u, v) and ∧ = ∧X is the product structure on the ringTCH1(R, 1;m) as in Theorem 7.1.Step 1. First, consider the case when u = v = 1, i.e., we prove Γa,1 ∧ Γb,1 =Γab,1. Recall that ∧ is defined as the composition ∆∗ µ∗ × in

X × A1 ×X × A

1 µ→ X ×X × A

1 ∆← X × A

1.

Under the identification X × X ≃ Spec (R ⊗k R), we have µ∗(Γa,1 × Γb,1) =Γ(a⊗1)(1⊗b),1, and ∆∗(Γ(a⊗1)(1⊗b),1) = Γab,1, because ∆ is given by the multi-plication R ⊗k R→ R. This proves (7.3) for Step 1.For the following remaining two steps, we use the projection formula: x ∧Vs(y) = Vs(Fs(x) ∧ y), which we can use by Theorem 7.1.Step 2. Consider the case when v = 1, but u ≥ 1 is any integer. We apply theprojection formula to x = Γb,1 and y = Γa,1 with s = u. Since TCH1(R, 1;m)is a commutative ring, by the projection formula, we get Vu(Γa,1) ∧ Γb,1 =Vu(Γa,1 ∧ Fu(Γb,1)). Here, the left hand side is Γa,u ∧ Γb,1 by eqn:FV identity,while the right hand side is =1 Vu(Γa,1 ∧Γbu,1) =2 Vu(Γabu,1) =

3 Γabu,u, where=1 and =3 hold by (7.2) and =2 holds by Step 1. This proves (7.3) for Step 2.Step 3. Finally, let u, v ≥ 1 be any integers. Let w = gcd(u, v). We againapply the projection formula to x = Vu(Γa,1), y = Γb,1, s = v, so that Vu(Γa,1)∧



Vv(Γb,1) = Vv(Fv(Vu(Γa,1))∧Γb,1). Its left hand side coincides with that of (7.3)by (7.2). Its right hand side is =1 Vv(Fv(Γa,u) ∧ Γb,1) =

2 Vv(wΓavw , uw∧ Γb,1),

where =1 and =2 hold by (7.2). But, Step 2 says that Γavw , uw∧Γb,1 = Γ

avw b

uw , uw

so that Vv(wΓavw , uw∧ Γb,1) = wVv(Γa

vw b

uw , uw

) =† wΓavw b

uw ,uvw

, where = † holds

by (7.2). This last expression is the right hand side of (7.3). Thus, we obtainthe equality (7.3) and this finishes the proof.

Theorem 7.10. For Spec (R) ∈ SmAffessk , TCH(R) is a restricted Witt-

complex over R, and its sub-pro-system TCHM (R) is a restricted sub-Witt-complex over R.

Proof. As saw in the proof of Theorem 7.1, we already have the restriction R,the differential δ, the Frobenius Fr and the Verschiebung Vr defined by the sameformulas. Furthermore, by Proposition 7.9, now we have ring homomorphismsλ = τR : Wm(R) → TCH1(R, 1;m) for m ≥ 1. The properties (i), (ii), (iii),(iv) in Section 2.2.2 are independent of the choice of the ring, so that what wechecked in Theorem 7.1 still work. To prove the theorem, the only thing leftto be checked is the property (v) that for all a ∈ R and r ≥ 1,

(7.4) FrδτR([a]) = τR([a]r−1)δτR([a]),

where we have shrunk the product notation ∧ and taken the ring homomor-phism λ to be τR. To check this, we identify Wm(R) with (1 + tR[[t]])×/(1 +tm+1R[[t]])×.If a = 0, then τR([a]) = Γ(1−0·t) = ∅. So, both sides of (7.4) are zero.If a = 1, then τR([a]) = τR(1 − t) = Γ(1−t). But, in our definition of δ, tocompute it, we should first restrict the cycle Γ(1−t) ⊂ Spec (R) × Gm ontoSpec (R) × (Gm \ 1), which becomes empty. Hence, δτR([a]) = δΓ(1−t) = 0,so again both sides of (7.4) are zero.Let a ∈ R \ 0, 1. Then τR([a]) = Γ(1−at) ⊂ Spec (R) × A1, and δτR([a]) isgiven by the ideal (1 − at, 1 − ty1) in R[t, y1]. Since t is not a zero-divisor inR[t, y1], we have (1− at, 1− ty1) = (1− at, y1− a) as ideals. Hence, FrδτR([a])is given by the ideal (1− art, y1 − a) in R[t, y1]. On the other hand,

(7.5)τR([a]

r−1)δτR([a]) = Γ(1−ar−1t) ∧ Spec(

R[t,y1](1−at,y1−a)

)

= ∆∗(

(R⊗kR)[t,y1](1−(ar−1⊗1)(1⊗a),y1−(1⊗a))

)=† Spec

(R[t,y1]

(1−art,y1−a)

),

where † holds because ∆ is induced by the product homomorphismR⊗kR→ R.Hence, both hand sides of (7.4) coincide. This completes the proof.

Theorem 7.11. For Spec (R) ∈ SmAffessk and n,m ≥ 1, there is a unique

homomorphism τRn,m : WmΩn−1R → TCHn(R, n;m) that defines a morphism of

restricted Witt-complexes over R, τR•,m : WmΩ•−1R → TCH•(R, •;m)m, such

that τR1,m = τR.

Proof. The theorem follows from Theorem 7.10 and [28, Proposition 1.15]. Wehave τR1,m = τR because the map λ of §2.2.2 is given by τR in Theorem 7.10.



We have shown in Propositions 7.6 and 7.9 that τR is a group homomorphismfor any k-algebra R and is a ring homomorphism if R is smooth. Here, weprovide the following information on τR.

Theorem 7.12. Let R be an integral domain which is an essentially of finitetype k-algebra. Then τR is injective. It is an isomorphism if R is a UFD.

Proof. Let K := Frac(R) and ι : R → K be the inclusion. This induces acommutative diagram

Wm(R)Wm(ι)

//

τR

Wm(K)

≃τK

TCH1(R, 1;m) // TCH1(K, 1;m),

where the bottom map is the flat pull-back via Spec (K) → Spec (R), and τKis the isomorphism by [28, Corollary 3.7]. Since Wm(ι) is clearly injective (see[28, Properties A.1.(i)]), it follows that τR is injective, too.Suppose now R is a UFD and V is an irreducible admissible cycle inTz1(R, 1;m). Then we must have (I(V ), t) = R[t], where I(V ) is the idealof V . Since R[t] is a UFD, using basic commutative algebra, one checks thatI(V ) = (1− tf(t)) for some non-zero polynomial f(t) ∈ R[t]. In particular, themap τR is surjective and hence an isomorphism.

7.3. Etale descent. Finally:

Proof of Theorem 1.4. By Corollary 5.15, we can assume |m| ≥ 2. We setY = X/G, λ = |G| and consider the diagram

(7.6) G×Xγ//

p

X

f

Xf

// Y,

where γ is the action map and p is the projection. Since G acts freely on X ,this square is Cartesian and f is etale of degree λ. By [6, Proposition 1.7], wehave f∗ f∗ = p∗ γ∗ : CHq(X [r]|Dm, n)→ CHq(X [r]|Dm, n).Since f is G-equivariant with respect to the trivial G-action on Y , we see thatf∗ induces a map f∗ : CHq(Y [r]|Dm, n) → CHq(X [r]|Dm, n)

G. Moreover, itfollows from [21, Theorem 3.12] that f∗ f

∗ is multiplication by λ.On the other hand, it follows easily from the action map γ that p∗ γ∗(α) =∑g∈G

g∗(α). In particular, p∗ γ∗(α) = λ · α if α ∈ CHq(X [r]|Dm, n)G.

Since λ ∈ k× and the Teichmuller map is multiplicative with |m| ≥ 2, wesee that λ ∈ (W(|m|−1)(k))

×. We conclude from Theorem 5.4(3) and Corol-

lary 5.14 that the composite CHq(Y [r]|Dm, n)f∗

−→ CHq(X [r]|Dm, n)G λ−1f∗−−−−→

CHq(Y [r]|Dm, n) yields the desired isomorphism.



Acknowledgments. The authors thank Wataru Kai for letting them know abouthis moving lemma for affine schemes. The authors also feel very grateful to thereferee whose careful and detailed comments had improved the paper and theeditor of the Documenta Mathematica for various kind suggestions. Duringthis research, JP was partially supported by the National Research Foundationof Korea (NRF) grants No. 2013042157 and No. 2015R1A2A2A01004120,Korea Institute for Advanced Study (KIAS) grant, all funded by the Koreangovernment (MSIP), and TJ Park Junior Faculty Fellowship funded by POSCOTJ Park Foundation.

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Amalendu KrishnaSchool of MathematicsTata Institute ofFundamental Research1 Homi Bhabha RoadColaba, Mumbai, [email protected]

Jinhyun ParkDepartment of Mathematical SciencesKAIST 291 Daehak-ro Yuseong-guDaejeon 34141Republic of Korea (South)[email protected]@kaist.edu


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On Additive Higher Chow Groups of Affine Schemes · 2016-01-23 · On Additive Higher Chow Groups of Affine Schemes Amalendu Krishna and Jinhyun Park Received: July 5, 2015 Revised: